Sequences and Series

• Adding Arithmetic Sequences [Tanner, 07/10/1998]
  How do you add the numbers from 1 to 5000 without actually doing it or using a calculator? What if you were adding just the odd numbers?

• Calculating the Fibonacci Sequence [Suhendra, 11/28/1996]
  Is there a formula to calculate the nth Fibonacci number?

• Coal Consumption [Plascencia, 01/13/1997]
  How long will coal reserves last if consumption increase at the rate of 3.1 percent per year?

• Decimal To Fraction Conversion [Lawson, 06/25/1998]
  I am trying to find a method (one that can be programmed on a PC) to convert the decimal part of a real number to a fraction represented by integers for the numerator and denominator.

• Doubling Grains of Wheat [Drumm, 10/7/1996]
  A man asked for 1 grain of wheat for the 1st square on a chess board, 2 grains for the 2nd square...

• Doubling Sequence [Maureen, 8/24/1996]
  On Jan 1st it snowed one centimeter; on Jan 2, 2cm; on Jan 3, 4 cm...

• Fibonacci and Incoming Bits [Suja, 09/08/99]
  Given a transmitter sending 100 bits of random data over an ideal communication channel, what is the probability that there will be three consecutive 1's at least once in the sequence?

• Finding a Pattern [Ray, 11/11/2001]
  Give the next four numbers in the sequence: 2, 8, 7, 28.

• Finding the Pattern in a Series of Numbers [Haule, 11/14/1995]
  What is the pattern for 1, 8, 27...?

• Finding sum formula using sequences of differences [Kyungsoo, 06/28/1998]
  Finding a formula for the sum of the first n fourth powers using sequences of differences.

• Infinite square root [Medalves, 6/4/1996]
  If y= sqrt(2+ sqrt(2+ sqrt(2+ sqrt(2+ ..., y=2,... how can I prove that this is true, using normal properties of roots?

• Look-and-Say Sequence [Vanessa, 02/14/2002]
  I can't find the next six numbers: 1, 11, 21, 1211, 111221, ...

• Mean Proportionals and Geometric Means [Mary, 01/06/1999]
  How do you find the mean proportional of two numbers? What about two mean proportionals? n mean proportionals?

• Next Number in a Sequence [KJS, 03/13/2002]
  Given any sequence, one can construct an infinite number of n-degree polynomials that satisfy the sequence, hence discern an infinite number of answers. What is the proof for this?

• Strategies for Tests on Sequences [Joel, 7/9/1996]
  I have a problem answering test questions about number sequences.

• Sum of n Odd Numbers [Mucha, 7/11/1996]
  Why is the sum of the first n odd numbers the square of n?

• The Traveling Bee [Emerling, 09/18/1998]
  If a bee travels between two trains that are moving at 30 and 20 mi/hr respectively, starting from 50 mi apart, how far does the bee travel?

• Unsolvable Equations [Shawn, 11/10/2001]
  If I have an equation in the form of x^n+y^n=z, how do I solve for n?

• Why is Zero the Limit? [Katie, 02/25/2002]
  Why is zero called the limit of the terms in the sequence the limit of 1 over n, as n approaches infinity, equals zero?

• Taylor series [Sledd, 11/3/1994]
  Please describe the Taylor series.

• Sequence of Triangular Numbers [Rudolph, 7/13/1995]
  What is the sequence called 1, 3, 6, 10, 15 and how is it generated?

• Finding the Missing Numbers in a Sequence [Tan, 11/30/1995]
  Fill in the blanks for this series of numbers based on its underlying pattern: 3, 4, 6, 8, 12, (), 18, 20, (), 30, 32

• Solving a Sequence [D'Andrea, 12/7/1995]
  I am teaching a 7th grade Pre-Algebra class and recently came across the following problem: Write an expression to find the nth term of the following sequence 3, 9, 18, 30, 45 . . .

• Finding a Term of an Arithmetic Series [Loyola, 12/13/1995]
  The fifth term of an arithmetic series is 16 and the sum of the first 10 terms is 145. Write the first three terms.

• Finding the 1000th Term in a Sequence [Muller, 1/19/1996]
  Two kids on a car trip decide to count telephone poles. One kid counts normally, 1,2,3,4,5...25,26,27...31,32,33, etc. The other kid counts them a different way: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1...

• Calculating a Series [Cams, 1/23/1996]
  I have a question about finding the formula that will calculate a series of numbers not starting with 1 and not necessarily increasing by 1...

• A Simple Expression? [Hellings, 1/26/1996]
  Is it possible to have a simple expression for a certain series starting at k=b...?

• Finding an Unknown Sequence [Sylvester, 3/31/1996]
  I can't figure out where to start with this Series and Sequences question: 1+3x+6(x)(x)+10(x)(x)(x)+15(x)(x)(x)(x)+. . .

• Sigma Notation [Lui, 4/14/1996]
  I am trying to find questions regarding sigma notation.

• Sum of Harmonic Series [Tarantola, 5/9/1996]
  What's the total sum in terms of variables for the series (1/1*2)+(2/2*3)......+n/n(n+1)?

• Arithmetic Series [Balchin, 5/19/1996]
  How do you calculate a series like 2,4,6,8... for say 3 terms starting anywhere in the series not by adding 3 specific terms together, but by using the first term and the number 3?

• Prove a Formula [Fox, 6/1/1996]
  1/2.3 + 1/3.4 + 1/4.5 +...+ n

• Common Ratio of a Geometric Progression [Cstse, 6/6/1996]
  Can the common ratio of a geometric progression be equal to 1 OR -1?

• Taylor approximation of tan^2(x) [Stribblehill, 6/12/1996]
  Just to check that I can't do this because f'(0) = infinity...

• Binomials and Products [Medalves, 6/21/1996]
  If S=1*2*3 + 2*3*4 + 3*4*5 + 4*5*6+...+48*49*50, how can I represent the answer of S by using a Binomial of Newton - n!/k!(n-k)! ?

• Series Problem: Find the Sum [Brasher, 6/24/1996]
  Find sum[sin(nx)/(3^n),(n,0,oo)] if sin x=1/3 and x is in the first quadrant.

• Infinite Sums [Gerard, 6/26/1996]
  Given the function f(x)= x^2/(1+x^2), find the sum...

• Two Arithmetic Series [Novica, 7/1/1996]
  an and bn are two arithmetic series; An and Bn are sums of first n elements. If (4n+27) * An = (7n+1) * Bn, compute a11/b11.

• Proof by Induction [Turner, 7/3/1996]
  How can I prove through induction that 1+1/4+1/9+ ... 1/n^2 < 2-1/n for all n > 1?

• Feeding Chickens - Arithmetical Progression [Siong, 7/6/1996]
  A farmer has 3000 hens. Each week he sells 20... what is the total cost of feeding the hens...?

• Arithmetical Progression [Siong, 7/7/1996]
  An arithmetical progression has a common difference of 1/1/2. The sum of the first n terms is 365 and the sum of the first 2n terms is 1330. Calculate the value of n and the first term.

• Circle in n Sectors [Siong, 7/8/1996]
  A circle is completely divided into n sectors in such a way that the angles of the sectors are in arithmetic progression...

• Pattern of Remainders [Sprenger, 7/10/1996]
  What is the pattern for numbers that have a remainder n-1 when divided by n for all n between 2 and a given upper bound?

• Sum of First n Odd Numbers [Siong, 7/10/1996]
  Show that the sum of the first n odd numbers is a perfect square.

• Sequence Pattern and Closed Form [Gillon, 7/16/1996]
  Given the pattern for a sequence, I can't figure out a general rule for the nth term.

• How to Find Patterns of Sequences [Joel, 7/16/1996]
  What are the patterns of the sequences 2,3,1,2,8,9 and 6,10,15,23,31,41?

• Unique Subset of Set of Fractions [Peters, 7/19/1996]
  How can I determine a set of fractions such that if I add any subset of those fractions, I get a result that is unique relative to the result of any other subset in this set?

• Sequences [Suryadie, 7/29/1996]
  The sum of three numbers is 147 and when multiplied together they yield 21952... Find a formula for 60, 30, 20, 15...

• Geometric Series and Sequences [Melanie, 8/9/1996]
  What term of the geometric sequence 3,6,12... is equal to 768?...For what range of values for y will the series {y+y^2+y^3+....y^n} have a limiting sum?...

• Test for Convergence [f16a10, 8/20/1996]
  Sum {from k=0 to infinity} [{log(k+1)-log k}/tan^(-1) (2/k)]

• Halving and Halving Again - Zeno's Paradox [Chris, 8/22/1996]
  Since we can never really get to zero by reducing something by halves, does that mean that we are floating on air?

• Proof that a Sequence Converges [Horrocks, 8/23/1996]
  Prove that, if | a | < 2 for all i = 1,2,3,..n, ...

• Rule for a Sequence [Michelle, 8/25/1996]
  What is the next number of these two sequences? {8 13 5 15 20 12} and {20 25 17 51 56 48}

• Predicting the Next Number [Smith, 8/30/1996]
  When given a series of numbers and asked to predict the next number, what is the formula for doing so?

• Sum of a series [Jang, 10/26/1996]
  Compute the sum of the coefficients of the expansion of (x+0.5)^100 for which the exponent is divisible by three.

• Level of Medicine in the Human Body [Brenchi, 11/1/1996]
  A patient receives a 10mg dose of medicine every four hours... prove that there will always be less than 40mg in the patient's body."

• Proof that INT(1/x)dx = lnx [Faulkner, 11/08/1996]
  How do you integrate (1/x)dx?

• A Monster of a Continued Fraction [Beebe, 11/09/1996]
  How do you find the value of an continued fraction?

• Volume of Water in an Urn [McElroy, 11/12/1996]
  An urn contains 1 liter of water; a second urn nearby is empty. After pouring the water back and forth 1,200,000 times in a certain way, how much water is left in each urn?

• Doubling Pennies [Valerie, 11/26/1996]
  If I start with a penny and double it daily for 30 days, how many pennies do I have at the end?

• Arithmetic Progression [Sushant, 12/19/1996]
  If (b+c-a)/a, (c+a-b)/b and (a+b-c)/c are in arithmetic progression, show that 1/a, 1/b and 1/c are also in arithmetic progression.

• Telescoping Series [Connie, 12/30/1996]
  Find the sum (to the nth term) of: 1/(1x3) + 1/(3x5) + 1/(5x7) +....+ 1/{(2n-1)(2n+1)}

• Recursive vs. Explicit Formulas [Johnsen, 01/02/1997]
  What is the difference between explicit and recursive formulas?

• Alternating Sequence [McKenzie, 01/27/1997]
  Find a pattern and the next three numbers in the sequence: 0, 8, 27...

• Rat Population [Hilton, 03/14/1997]
  Two rats have 6 offspring, 3 of which are female. Each female reaches maturity at 120 days and produces a litter of 6 every 40 days thereafter. How many rats will there be in a year?

• Geometric Sequence? [Gemilang, 03/21/1997]
  Is {1, -1, 1, -1 ...} a geometric sequence?

• Triangular Numbers in a Proof [Flanagan, 04/08/1997]
  How do you prove 1^3+2^3+3^3+ ...+n^3 = (1+2+3+...+n)^2 by induction?

• Series Types [Sam, 05/11/1997]
  What are the definitions of convergent, divergent, and oscillating series?

• Program to Calculate Pi [Kirklin, 05/23/1997]
  I am trying to write a program on my TI-83 calculator to calculate the infinite digits of Pi while displaying them on-screen.

• Proof of Series Sum [Piggy, 06/19/1997]
  Prove that 1x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 ... +n(n+1) = (nx(n+1) x (n+2))/3.

• Continued Fractions [MacDonell, 07/02/1997]
  What is a continued fraction?

• Counting School Supplies for the First Days of School [Bender, 07/23/1997]
  The rule is n(n+1)/2 for each day and n(n+1)(n+2)/6 for the entire sequence. Where did the divide by 2 and 6 come from?

• Multiplying Mice [Slavic, 07/23/1997]
  Baby mice can breed when they are 6 weeks old and the babies are born after 3 weeks. If each mother mouse has only one litter and all the litters have 8 babies, half males and half females, how many mice will you have 18 weeks from today?

• What is the nth Term? [Chris, 08/02/1997]
  The first four terms of a sequence are 16, 8, 4, 2. Find the next two terms and the rule for the nth term.

• Number Sequence Problem [Mann, 08/11/1997]
  I have a number sequence but can not find out the pattern.

• Probability of Random Numbers Being Coprime [Knobler, 08/12/1997]
  I have heard that the probability of two randomly selected integers being coprime is 6/(pi^2). How do you show this is true?

• Integer Sequence [Stang, 08/15/1997]
  Show that if 19 distinct integers are chosen from the sequence 1,4,7,10,13,16,19...,97,100, there must be two whose sum is 104.

• Limits of Sequences [Mooney, 08/19/1997]
  Please explain the limit superior of a sequence .

• Nth Term of a Series [Conrad, 08/27/1997]
  1/(1*2*3)+1/(2*3*4)+.......+1/+.......+1/(100*101*102) =?

• 21^100 - Last Two Digits [Yougo, 09/04/1997]
  What are the last two digits of 21 to the 100th power?

• Prize Money [Vickers, 09/04/1997]
  If first prize wins $1,000 out of $6,000 and twentieth prize wins $100, how much money do second through nineteenth place win? Is this a geometric sequence?

• Continued Fraction [MacDonell, 09/18/1997]
  How would you express sqrt3 - 1 as a continued fraction?

• Advanced Algebra [Bailey, 09/23/1997]
  My teacher gave us this problem: 1+1/(1+1/(1+1/(1+1/1+...)))

• Square Root of 3 minus 1 [Lam, 09/24/1997]
  Express sqrt3-1 as a continued fraction.

• Infinite Sequence [Leone, 09/28/1997]
  From f:n ---> 3 + (-1/2)^n where n belongs to the set of natural numbers, describe a strip that contains all but a finite number of points of the graph of f.

• Continued Fractions [Gold, 10/07/1997]
  Exactly what are "continued fractions"?

• Limits - Indeterminate Forms [Mary, 10/12/1997]
  I cannot do a problem where I need to convert into the form 0/0 and then use L'Hopital's Rule...

• Alternating Harmonic Series [Campbell, 11/18/1997]
  I am trying to find the proof for the sum of the alternating harmonic series. I did find out that it is ln(2), but please tell me why?

• Infinite Product [Nguyen, 12/01/1997]
  How do I find the infinite product of 3^1/3 * 9^1/9 * 27^1/27 * ..... (3^n)^1/3^n ?

• Infinite Series Involving Pi [Tak, 12/16/1997]
  I need a reason why 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ..... = pi^2/6 is correct. What if this series alternates?

• Natural Numbers [Moon, 01/08/1998]
  What are two ways of finding the sum of n natural numbers?

• Infinite Series of 1/n [Theemling, 01/13/1998]
  What is the sum of an infinite series of 1/n when n = 1,2,3...? I understand the answer is divergence or the sum is infinity, but not why, especially since the terms eventually go to 0.

• Completing Geometric Sequences [Caudill, 02/16/1998]
  How do you find the missing terms of a geometric sequence?

• The Limit of (1+1/x)^x As x Approaches Infinity [Harvey, 02/17/1998]
  How Euler calculated e, and what it has to do with the equation (1+1/x)^x.

• Formula For the Sum Of the First N Squares [Sandin, 02/20/1998]
  Can you show me how (1^2+2^2+3^2 +...+N^2) becomes (N*(N+1) * (2N+1))/ 6?

• Convergence of an Alternating Series [Welt, 02/23/1998]
  Why does the alternating series (-1)^(n+1)*(ln (n)/n) converge?

• Limit of Area [Jason, 03/01/1998]
  Limit approached by area of a square when its sides are repeatedly divided into three congruent parts and squares are constructed outwardly on the middle parts.

• What is a continued fraction? [Wollak, 03/06/1998]
  What is a continued fraction and what makes it different from the types of fractions or ratios I'm used to?

• Changing a Recurrence Relation into a General Formula [Wollacott, 03/06/1998]
  Expressing as a general formula a recurrence relation similar to that for the Fibonacci sequence.

• Proving Series Convergence [Wolfson, 03/08/1998]
  Show that the infinite series 1/a(n) converges, where a(n) are the positive integers that do not contain a 2.

• Limit of (-1)^n? [Bourque, 03/14/1998]
  As n approaches infinity, does (-1)^n have a limit?

• Arithmetic and Geometric Progressions [Angol, 03/23/1998]
  Given a set of conditions, can you find a specific term in an arithmetic or geometric progression?

• e as a Series and a Limit [Lee, 03/30/1998]
  Why does e = 1 + 1/2! + 1/3! + 1/4! + ... and lim (1 + 1/n) ^ n, as n --> infinity?

• 22/7 as an Approximation for Pi [Faulkner, 04/01/1998]
  Approximating pi by simple continued fractions.

• Two Ways to Find a Formula [Mon, 04/14/1998]
  I need to show that Sigma(rx^r) = (x-(n+1)x^(n+1)+nx^(n+2))/(1-x)^2.

• Counting Regions Formed by Straight Lines [Shah, 04/18/1998]
  How many regions are formed by n straight lines if no three meet in a single point and no two are parallel?

• Summing Triangle Numbers [Hooton, 04/21/1998]
  Can you help me find the formula to find the sum of a finite number of triangle numbers?

• Investigating Sequence Patterns [Condo, 05/14/1998]
  How can we find the pattern in the following sequence? Take a circle with three dots on the circumference and connect with lines...

• Cockroach Traveling Along an Elastic Tightrope [Fields, 05/15/1998]
  Finding the harmonic series in a problem of related rates.

• Counting Regions Formed by Chords of a Circle [Shah, 05/19/1998]
  Determining the number of regions formed by connecting n points on the circumference of a circle.

• Equation of a Sequence with Constant Third Differences [Malik, 05/26/1998]
  Using the method of difference or the Gregory-Newton formula.

• A Fibonacci Proof by Induction [Wang, 06/05/1998]
  Let u_1, u_2, ... be the Fibonacci sequence. Prove by induction...

• Rational Series That Sum to an Irrational Number [Carvajal, 07/06/1998]
  How can the sum of an infinite series of rational numbers result in an irrational number?

• Sum of An Infinite Series [May, 07/08/1998]
  Is it possible to add up all the terms of an infinite series?

• Convergent and Divergent Series [Gena, 07/14/1998]
  I cannot seem to solve these problems...

• Summing a Binary Function Sequence [Ping, 07/16/1998]
  How do you compute the sum of B(n)/(n(n+1)) from 1 to infinity, where B(n) denotes the sum of the binary digits of n?

• Nested Square Roots [Natasha, 07/17/1998]
  Solve for n where n = sqrt(6 + sqrt(6 + sqrt6 + ...

• Sequence of Squares [Sam, 07/25/1998]
  Do you have any information on the sequence of squares?

• Series Expansion of 1/(1-x) [Edic, 08/01/1998]
  Can you explain the series expansion identity 1/(1-x) = 1 + x + x^2 + x^3 + ... ? In what region does it converge?

• Using Sine and Tangent to Find Pi [Marshall, 08/01/1998]
  I want to know how to determine the sine or tangent of an angle without using a calculator.

• Subtracting Finite Sums of Integers [Wilson, 08/03/1998]
  If n = 1 + 3 + 5 + 7 + ... + 999 and m = 2 + 4 + 6 + 8 + ... + 1000, what does m-n equal?

• Irrational Decimals [Bennet, 08/04/1998]
  I know the proof stating that 0.9 (repeating) is actually equal to one, but from a representation standpoint are they actually considered to be mathematically equal?

• Solving Continued Fractions [Kristy, 08/08/1998]
  How do you get sqrt(2) from 1/(2 + 1/(2 + 1/(2 + ...)))? How do you solve continuous fractions in general?

• Summing an Oscillating Series [Anonymous, 08/10/1998]
  Does 1 - 1 + 1 - 1 + 1 - ... equal 1 or 0

• Fourier Series and the Zeta Function [Perry, 08/13/1998]
  How do you evaluate Zeta of 2?

• Formula for a Sequence [Alvear, 08/21/1998]
  A variety of techniques for solving problems like this one.

• Summing Consecutive Integers [Simon, 08/30/1998]
  Express 1994 as a sum of consecutive positive integers, and show that this is the only way to do it.

• Power Series from Long Division [Sam, 08/31/1998]
  How can you use long division of polynomials to get the power series expansion of 1/(1-x)^2?

• Summing Integers to the Fourth Power [Ritter, 09/26/1998]
  How do you find the formula for the sum of integers to the fourth power: 1^4 + 2^4 + ... +n^4?

• Activities to Find Pi [Peuler, 10/07/1998]
  Can you suggest any classroom activities to find pi, other than the standard way of measuring the circumferences and diameters of circles?

• The Origin of Lucas Numbers [Smith, 10/08/1998]
  I need help with Lucas Numbers - how and why they were created.

• From Infinite Decimals to Mixed Fractions [Jessi, 10/09/1998]
  Write this fraction as a mixed number: (.151515...+.555...)/ (.161616...-.2222...).

• Factors and Multiples - Hamiltonian Path [Dawson, 11/02/1998]
  We have to make a sequence of numbers, all different, each of which is a factor or a multiple of the one preceding it.

• Defining a Sequence [Bell, 11/04/1998]
  Can you define sequence, series, convergence, and divergence, and explain how they correlate to one another?

• Finding the Digit of a Decimal Expansion [Breanna, 11/14/1998]
  What digit will appear in the 534th place after the decimal point in the decimal representation of 5/13?

• Working with Sequences [Kwan, 11/14/1998]
  Give the next two terms of the sequence: 1, 1, 2, 4, 3, 9, ....

• A Hundred-Row Number Pyramid [Williams, 11/19/1998]
  Starting with two(1,2) in the first row of a pyramid and adding one more as you go down the list, what is the last number on the righthand side in the 100th row?

• Figurate and Polygonal Numbers [Megan, 11/21/1998]
  I need to know everything about figurate numbers.

• Summing n^k [Kijjaz, 11/24/1998]
  Is there a general formula for summing the n^k, where k is a positive integer?

• Counting Bug Populations [Ahmed, 12/03/1998]
  In each generation, a happy bug splits into a sad bug and a blank bug, .... How do you find a formula for the number of each kind of bug in generation n?

• Recursive and Explicit Formulas [Ososke, 01/19/1999]
  Is there an easy way to convert recursive formulas into explicit ones and vice versa?

• Tricks to Sum an Infinite Series [Heggie, 01/23/1999]
  What does the sum of [(-1)^n*|sin n|]/n from n = 1 to infinity converge to?

• Gauss' Formula [Roxanne, 02/03/1999]
  Does Gauss' formula work for any progression or only for arithmetic progressions?

• Writing Sigma Notation [Mary, 02/03/1999]
  What does the equation that goes under the Sigma notation imply?

• Laurent Expansion [Genni, 02/25/1999]
  Explanation the Laurent Expansion using some examples.

• Why Study Sequences and Series? [Tara, 02/26/1999]
  What are some applications of arithmetic sequences and series?

• Series Divergence [Ellen, 03/03/1999]
  Show that the series sum(k=0 -> infinity): (k/e)^k/k! is divergent.

• Declining Balance Interest [Maria, 03/22/1999]
  Can you explain declining balance interest to a high school business class?

• What are Taylor series? [Springer, 04/18/1999]
  What are Taylor Series? What are they needed for?

• Repeating Decimals [Grabowski, 04/28/1999]
  I am interested in finding longer repeating groups in number tails of repeating decimals.

• Block Tower [Koos, 05/05/1999]
  If a tower has a center with 6 blocks and four adjacent wings with blocks in descending order (5, 4, 3, 2, 1), how many blocks are there?

• Cutting a Square [Joubert, 05/05/1999]
  What is the maximum number of pieces that a square can be cut into with 70 straight cuts?

• Convergence of Sums [Andrews, 05/07/1999]
  Deduce that the following sum converges absolutely: Sum from 1 to infinity of (-1)^(n-1)/(n^2)!

• Fibonacci Sequence - An Example [Chow, 05/12/1999]
  Glass plates and reflections.

• Let f(x) = 1 + 1/2 + 1/3 + ... + 1/[(2^n)-1] [Manghat, 05/15/1999]
  Which of the following inequalities are correct?

• Sum of Inverse of Primes [Puertas, 05/25/1999]
  Is the infinite series S = 1/1 + 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + ... + 1/p(n) + ... convergent or divergent?

• Finding Number Patterns [Thorsheim, 05/29/1999]
  I am trying to find the pattern of the numbers 3,8,13,18,23,28,30,31,32...

• Ramsey's Theorem and Infinite Sequence [Chan, 06/01/1999]
  Ramsey's Theorem applied to divisibility in infinite sequences.

• Ant Walking in a Squared Spiral [Marnell, 06/02/1999]
  An ant walks out a distance of 1 from the origin, down the x-axis. It then turns left and goes up 1/2. If it continues turning left and going the half the previous distance, where does the ant end up?

• Expansion of (x+y)^(1/2) [Dheera, 06/07/1999]
  Is there a way to expand (x+y)^(1/2)? If so, how is it derived?

• Sum of a Sequence [Bethany, 06/10/1999]
  How would we find the sum of the sequence (3,4,6,9,13, ..., 499503)?

• Finding a Non-Recursive Formula [Alume, 06/10/1999]
  How can I find a non-recursive formula for the recurrence relation s_n = - [s_(n-1)] - n^2 with the initial condition s_0 = 3?

• Finite Differences and Polynomials [Harris, 06/17/1999]
  Can you show me why the differences between terms of a polynomial of degree n go constant at the nth row of differences?

• Cubes in a Grid [Holman, 06/17/1999]
  How can I find a formula for determining the number of cubes if a 3x3 grid requires 7 cubes, a 5x5 needs 25, a 7x7 needs 63, and so on?

• Triangle Series [Singh, 07/09/1999]
  What is the formula for finding the sum of the nth row of a triangle of numbers?

• Which Elements are Divisible by 11? [Knotts, 07/12/1999]
  Determine all solutions w = x+y+z, (1/w) = (1/x)+(1/y)+(1/z). In a given sequence, find which elements of the sequence are divisible by 11.

• Grains of Wheat [Khalafalla, 07/14/1999]
  The person who invented the game of chess was said to have been offered any payment he wanted... How much wheat did he receive?

• Co-efficient of an Algebraic Term [Elizabeth, 07/18/1999]
  Simplify [z^40] (1 + z + z^2 + ... + z^9)^100.

• Formula for the Nth Term in a Geometric Sequence [Andy, 08/05/1999]
  How can I tell whether the sequence a^2/2, a^4/4, a^6/8, ... is arithmetic or geometric, and how can I find a formula for the nth term?

• Interpolation and Extrapolation [Santiago, 08/10/1999]
  Can you give me a step-by-step procedure for finding missing values based on interpolation and extrapolation?

• Wrong Number in a Sequence [Bossingham, 08/13/1999]
  In the sequence {1, 1, 2, 2, 6, 18, 21, 84, 88, ...} which number is incorrect? What number does it need to be replaced with?

• Geometric Sequences and Series [JC, 09/10/99]
  How can I find the 13th and 20th terms of a sequence that begins -3072, 1536, -768, ...? How can I find the sum of the first 9 terms?

• Proof Using Pell's Equation [Nandung, 09/18/1999]
  Given Pell's equation for sqrt(D) and p/q = [a1;a2;...;an], can you prove that p^2 - D.q^2 = (-1)^n.k?

• Sum of Convergent Series [Abby, 09/25/1999]
  How can you find the sum for k = 0 to infinity of 1/[(k+1)(k+3)], and the sum for k = 0 to infinity of [(25/10^k) - (6/100^k)]?

• Finding a Series Given the Sum [Mike, 09/27/1999]
  How can I find all series of consecutive integers whose sum is a given value x?

• Newton's Method and Continued Fractions [Kaluhiokalani, 10/06/1999]
  Can you clarify some points on Newton's method of finding square roots without a calculator, and on the continued fraction algorithm (CFA)?

• Summing Odd Numbers Geometrically [Turkseven, 10/30/1999]
  Can you prove that 1 + 3 + 5 + ... + (2n-1) = n*n by using a simple geometric method?

• Catalan Numbers [Bradley, 12/15/1999]
  What are Catalan numbers and what applications do we have for them?

• Counting Squares in Bigger Squares [Julz, 02/29/2000]
  How many edge 2 squares (2x2 squares) can be found in an edge 4 square (a 4x4 square)?

• Euler's summmation of 1/n^2 [Bennett, 03/15/2000]
  Prove that pi^2/6 = the summation of 1/n^2 from 1 to infinity.

• Increasing and Decreasing Subsequences; Pigeonhole Principle [Elizabeth, 03/21/2000]
  How can I prove that there exists an increasing OR decreasing subsequence of length n+1 or more in any list of (n^2)+1 distinct integers?

• Exponential Generating Function [Kevin, 05/06/2000]
  How can I prove that the exponential generating function of the series 1, 1*3, 1*3*5, 1*3*5*7, ... is 1/sqrt(1-2*x)?

• Sum of Consecutive Cubes [Cuong, 05/11/2000]
  How can I prove that the sum of consecutive cubes equals a square? That is, 1^3 + 2^3 + 3^3 + ... + n^3 = m^2.

• Convergence of Alternating Series [Ki, 05/14/2000]
  Why is the test for convergence of alternating series a_n is greater than a_n+1 AND limit (an n goes to infinity) a_n = 0, instead of just limit (an n goes to infinity) a_n = 0?

• Finding an Explicit Formula for a Recursive Series [Phillips, 05/17/2000]
  How far will a man end up from his home if he walks a mile west, then walks east one half that distance, then walks west half of the distance he has just walked, and so on?

• Sum of Fibonacci Series [Barton, 05/23/2000]
  Is there a formula for the sum of the first n numbers in the Fibonacci sequence?

• Are All Infinitely Long Repeating Numbers Even? [Huggins, 06/06/2000]
  Given an infinitely long repeating series, x = 12341234..., then 10000x = 123412341234... Since 9999 is odd and 12340000... is even, can we say that x is even, and therefore all infinitely long repeating series are even?

• Using the Geometric Mean in a Sequence [Kari, 07/11/2000]
  How do you use the geometric mean to find the missing number in the geometric sequence 5,15,_,135,405?

• Sum of 1/n^2 [Schmidt, 07/24/2000]
  Computing the sum of 1/n^2 without using Fourier series.

• Summations of n^(-2k) [Rosel, 09/10/2000]
  How can I find the summations of the following series for n = 1 to infinity: (n^-2), (n^-4), (n^-[2k]) and (n^-[2k+1])?

• Proof of Convergence [Decroos, 09/29/2000]
  Why does the ratio F(n+1)/F(n) for the Fibonacci numbers converge to the golden ratio?

• Method of Finite Differences [Gillett, 10/12/2000]
  How can I find the generating equation for the series -3, 2, 13, 30, 53?

• Power Series for Sine and Cosine [Mark, 10/12/2000]
  Can you explain, without using calculus, how to get the power series for sine and cosine?

• Series for which Convergence is Unknown [Veronique, 11/09/2000]
  Are there series for which it is unknown whether they converge or diverge?

• Tribonacci Numbers [aznx, 11/11/2000]
  Is there an implicit formula to calculate the nth Tribonacci number? Also, is there a formula to find the sum of the first n Tribonacci numbers?

• Sum of Any Infinite Series [Keesing, 11/19/2000]
  Even if you can determine that a series converges, it's usually impossible to calculate the sum exactly. Why?

• Sum of 1/Sqrt(i) [Khoury, 11/20/2000]
  What is the formula for the sum of 1/sqrt(i) for i = 1 to n? Can you show me the proof by induction?

• Solutions to X^Y = Y^X [Pelz, 12/21/2000]
  How can I find the solutions to the equation x^y = y^x? I have been told that it involves the Lambert W Relation.

• Series Convergence [Chris, 01/27/2001]
  Test these series for convergence; if the series is alternating, tell whether the convergence is conditional or absolute...

• Sums of Consecutive Integers [Kasey, 02/04/2001]
  What numbers can be expressed as the sum of a string of consecutive positive integers?

• Sum of a Power Series [Ivan, 02/10/2001]
  How can I calculate the sum of the power series x + 4x^2 + 9x^3 + 16x^4 + ... + n^2x^n + ...?

• Repeating Decimals as Geometric Series [Stephanie, 02/13/2001]
  How can you express repeating decimals as geometric series and convert them to fractions using the series sum formula?

• Fourier Sine and Cosine Series [Bong, 02/23/2001]
  Why do we need Fourier Sine Series and Fourier Cosine Series? Each series has its own formulae. How do we know when to use them?

• Limits of Sequences [Kapoor, 02/25/2001]
  Is the limit of [(1 + 1/sqrt(n))^(1.5n)], as n goes to infinity, e? What is the limit as n goes to infinity of [(1 + a/n)^n], where a is not equal to 0?

• Millionth Digit of the Counting Numbers [Bukovich, 02/26/2001]
  A number is formed by writing the counting numbers in order: 123456789101112131415... What is the one millionth digit in this number?

• Stair Patterns [Aghajani, 02/27/2001]
  The 1st step is made with 4 matches, the 2nd with 10 matches, the 3rd with 18, the fourth with 28. How many matches would be needed to build 6, 10, and 50 steps?

• Series Convergence [Julian, 02/28/2001]
  Why does 1 + 1/2^z + 1/3^z + ... converge for Re(z) greater than 1?

• Sums of Consecutive Positive Integers [Loulie, 03/02/2001]
  Why are the powers of 2 are the only numbers you cannot get as the sum of a series of consecutive positive integers?

• Infinite Series [Klein, 03/12/2001]
  Given two jugs, each containing one litre of water, pour half of one jug into the other...

• One-to-One Correspondence of Infinite Sets [Anokye, 03/26/2001]
  How can I prove that any two infinite subsets of the natural numbers can be put in a 1-1 correspondence?

• Telescoping Series [Karen, 03/28/2001]
  Find the nth term in this pattern...

• Diagonal Sum in Pascal's Triangle [Potter, 04/02/2001]
  Find the sum of the reciprocals of the diagonals in Pascal's triangle.

• Trigonometric Equation for a Sequence [Gruber, 04/03/2001]
  I need an equation for the sequence 0, 0, 1, 0, 0, 1, 0, 0, 1, ... using the sine or cosine function.

• Mangoes at the Gates [Dickerson, 04/06/2001]
  To pick some mangoes from a tree inside seven walls with seven guards, you tell each guard that you'll give him half of the mangos you have, but he must give you back one mango. What's the minimum number of mangos you must pick to have at least one mango left?

• Millionth Term [Steve, 04/16/2001]
  What is the millionth term in the sequence 1, 2, 2, 3, 3, 3, ... ?

• Possible Proof That 1 + 1 Does Not Equal 2 [Desmond, 04/19/2001]
  I just want verification... would this proof work?

• Deriving Trig Functions; Taylor Series [Harold, 05/01/2001]
  How would I find, from first principles - no tables, no calculator - for example, 32 degrees? If I use a formula, how is it derived?

• Terms of the Series 1/n [Steer, 05/03/2001]
  How many terms of the series 1/1 + 1/2 + 1/3 + ... + 1/n would I need to guarantee that the sum will be larger than some given value x?

• Exponential Series Proof [Jake, 05/05/2001]
  Given e^x greater than or equal to 1 + x for all real values of x,and that (1+1)(1+(1/2))(1+(1/3))...(1+(1/n)) = n+1, prove that e^(1+(1/2)+ (1/3)+...+(1/n)) is greater than n. Also, find a value of n for which 1=(1/2)+(1/3)+...+(1/n) is greater than 100.

• Sum of Consecutive Squares [Robert, 05/11/2001]
  The sum for i = 1 to n, of i^2, is equal to ((n)(n+1)(2n+1))/6. Why?

• Non-Recursive Formula [Dang, 06/05/2001]
  I want to know the non-recursive formula of the nth number in the general Fibonacci sequence...

• Second-Order Linear Recurrences [Lanada, 06/08/2001]
  Three problems involving recurrence equations.

• Pyramidal Numbers [Karinny, 07/04/2001]
  Can I make a square pyramid with 1000 tennis balls?

• Differences Method [Mark, 07/06/2001]
  How does the method of differences work?

• Sum of Consecutive Odd Integers [Hooji, 07/27/2001]
  Given an integer N, can N can be written as a sum of consecutive odd integers? If so, how can I identify *all* the sets of consecutive odd integers that add up to N?

• Summation Notation and Arithmetic Series [Rachel, 07/27/2001]
  Do I need to use the arithmetic series formulas when doing sigma notation?

• Find the 10th Number [Adam, 08/10/2001]
  Find the 10th number in the sequence: 2 4 3 6 5 10.

• Find the 276th Letter [Cale, 08/15/2001]
  Find the letter that is the 276th entry in the following sequence: g,l,g,l,l,g,l,l,l,g,l,l,l,l,g,l,l,...

• Formula for Sum of First N squares [Mark, 08/23/2001]
  Can you show me how (1^2+2^2+3^2+...+N^2) becomes (N*(N+1)*(2N+1))/6 using a method other than a cubing pattern?

• Sigma Notation [Steve, 09/07/2001]
  To prove that sigma (i^2) from i = 1 to n i equal to (n(n+1)(2n+1))/6 start with (i+1)^3 - i^3...

• Counting Positive Rational Numbers [Alfred, 09/09/2001]
  In Hardy's book _Pure Mathematics_ he gives a formula for counting the positive rational numbers p/q when they are arranged in a triangular matrix and counted down diagonally from the top row... how can it be proved for all such numbers?

• Maclaurin Series for Tangent [Daniel, 09/17/2001]
  What is the Maclaurin series for tangent (not inverse tangent)?

• Pages in a Book [Patricia, 09/13/2001]
  A book is made of folded sheets of paper, each comprising four pages. One of the sheets has page numbers 88 and 169. How many pages are there in the book? What is the sum of all the page numbers in the book?

• Why Use Radians instead of Degrees? [Kanishka, 09/25/2001]
  Consider Maclaurin's theorem, from which we derive the Taylor series...

• Coding Pairs of Numbers [SoonMin, 10/18/2001]
  Using the equation: 1/2 ((a + b)^2 + 3a + b) I have plugged in numbers for a and b and worked it out, but I do not see how that "codes the pair" (a,b) into a single number.

• Math Virus Formula [Victor, 10/23/2001]
  The virus spreads to all the squares directly touching each other (not including diagonally) and I have found the formula for the number of newly infected cells (although this does not include the first minute)...

• 121, 111211, 311221 Puzzle [Joey, 10/23/2001]
  121, 111211, 311221 - what's the next number?

• Summing a Series Like n*(n!) [Aberlig, 10/28/2001]
  How can I add up a series like 1*1! + 2*2! + 3*3! ... n*n! ?

• Scoring System Problem [Kate, 10/28/2001]
  What is the highest score that is impossible to make?

• Looking for Patterns [Amanda, 10/30/2001]
  What would be the answer to: (x-a)(x-b)(x-c)(x-d)(x-e)...(x-y)(x-z)?

• Proof of Series ln(1+x) [Colleen, 11/15/2001]
  I need to show that the series ln(1+x) equals x-x^2/2+x^3/3-x^4/4, and so on, whenever x is between -1 and 1.

• Square Root Theory [Timothy, 11/16/2001]
  When I enter any positive number in the calculator or a fraction like 0.1, then take the square root of that number, then take the square root of that number, and keep pressing the square root button over and over, I eventually get to number 1. Why?

• Taylor Expansion [Henning, 11/21/2001]
  Can you give me the proof of this statement: arcsin(x) = x + 1/2 (x^3/ 3) + (1/2)(3/4)(x^5/5) + (1/2)(3/4)(5/6)(x^7/7) + ... The basis of the calculation is a Taylor series.

• Venn Diagram [Bryan, 11/25/2001]
  What does a five-set Venn diagram look like?

• Taylor Series Expansion [Don, 11/24/2001]
  A distance from A to B is 1000 meters. As one traverses it at 1 meter per second, the distance is instantaneously and uniformly stretched 1000 additional meters. How long does it take to get from A to B?

• Laying Paving Stones [Brian, 11/28/2001]
  Finding a relation for a sequence that relates to the number of ways paving stones can be laid to make a 3-foot-wide path using 3-foot by 1-foot stones.

• House of Cards [Jo, 12/02/2001]
  Is there a rule for working out the number of cards you need to build a house of cards of any size?

• Bit Strings with Even Numbers; Coin Toss [Erin, 12/09/2001]
  How many bit strings of length n have an even number of 1's? A fair coin is tossed until 2 consecutive heads appear. What's the probability that this will happen within the first n tosses?

• Infinite Series Problem [Flori, 01/15/2002]
  I have to find the correct answer to an infinite series question about three people throwing dice...

• Defining Quadratic Formula [Monica, 01/21/2002]
  Taking successive differences in a sequence.

• Find the Formula: 1, 6, 19, 44, 85 [Krisna, 01/31/2002]
  What are the steps to find the formula?

• Generalised 'Fibonacci' Series and Phi [Stuart, 02/10/2002]
  A Fibonacci-style series that starts with any two numbers and adds successive items produces a ratio of successive items that converges to phi in about the same number of terms as for the basic Fibonacci series. Is this well known and provable?

• The Sequence Sin(n) [Sean, 02/20/2002]
  I am trying to prove that the sequence sin(n), for n, a natural number, does not converge.

• Nested Radical [Mac, 02/25/2002]
  Prove the following nested radical: sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...)))) = 3

• Convergence of Euler's Infinite Series [Chris, 03/06/2002]
  I am writing to inquire about the convergence of the infinite series 1+(1/2^2)+(1/3^2)+(1/4^2)+(1/5^2)+(1/6^2)+...+(1/n^2) = ?

• Cubes in a Big Cube [Yannik, 03/11/2002]
  Is there a formula for the number of cubes in an n*n*n cube?

• Balance Point between Converging and Diverging Infinite Series [Jim, 03/20/2002]
  Is there a way to find the balance point between convergence and divergence for any type of series - that point where the n(th)term gets smaller JUST fast enough for the series to converge?

• Fraction Algorithm [Gil, 03/19/2002]
  I have been having trouble making an application that can convert a finite decimal to a fraction without doing 78349/1000000.

• Sum of i [Julie, 03/23/2002]
  If the sum as i goes from 1 to n of 2^i is 2^n -1, what is the sum as i goes from 1 to n of 3^i ?

• Fresnel Integral of sin(x^2) [Robert, 03/28/2002]
  I can not see how this integral from 0 to infinity can have any limit at all. If we used infinite series, we would not get a value for x = infinity.