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 TOPICS This page:   sequences/series    Search   Dr. Math See also the Dr. Math FAQ:   golden ratio,   Fibonacci sequence Internet Library:   sequences/sets HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Sequences, Series Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Strategies for finding sequences. Average Yearly Depreciation [06/07/2002] I'm trying to figure out the average depreciation per year for an automobile, given its price history over several years. Balance Point between Converging and Diverging Infinite Series [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? Binomials and Products [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)! ? Bit Strings with Even Numbers; Coin Toss [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? Block Tower [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? Calculating a Series [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... Card Stacking Problem [06/10/2006] Is it possible to stack a deck of cards so that the top card on the stack does not overlap the bottom card at all? Changing a Recurrence Relation into a General Formula [03/06/1998] Expressing as a general formula a recurrence relation similar to that for the Fibonacci sequence. Checkerboard Problem [07/30/1997] Is there an exponential pattern for how many squares there are on a checkerboard? Circle in n Sectors [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... Cockroach Traveling Along an Elastic Tightrope [05/15/1998] Finding the harmonic series in a problem of related rates. Coding Pairs of Numbers [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. Co-efficient of an Algebraic Term [07/18/1999] Simplify [z^40] (1 + z + z^2 + ... + z^9)^100. Common Ratio of a Geometric Progression [6/6/1996] Can the common ratio of a geometric progression be equal to 1 OR -1? Completing Geometric Sequences [02/16/1998] How do you find the missing terms of a geometric sequence? Computing e^(-2) Using a Power Series [04/16/2005] Use a power series to compute e^(-2) to four decimal place accuracy. Connell Sequence [06/12/2003] The sequence 1,2,4,5,7,9,10,12,14,16,17... has one odd number followed by two evens, then three odds, four evens, and so on. What number is the 2003th term? Continued Fraction [09/18/1997] How would you express sqrt3 - 1 as a continued fraction? Continued Fractions [10/07/1997] Exactly what are "continued fractions"? Continued Fractions [07/02/1997] What is a continued fraction? Convergence of Alternating Series [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? Convergence of an Alternating Series [02/23/1998] Why does the alternating series (-1)^(n+1)*(ln (n)/n) converge? Convergence of Euler's Infinite Series [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) = ? Convergence of Sums [05/07/1999] Deduce that the following sum converges absolutely: Sum from 1 to infinity of (-1)^(n-1)/(n^2)! Convergent and Divergent Series [07/14/1998] I cannot seem to solve these problems... Converting a Product Function to a Summation [07/24/2004] How do I convert the product of n terms to a summation? For example, if f(x) = a(x - a1)(x - a2)...(x - an), how do I get f(x) = the sum of some series? Counting Bug Populations [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? Counting Positive Rational Numbers [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? Counting Regions Formed by Chords of a Circle [05/19/1998] Determining the number of regions formed by connecting n points on the circumference of a circle. Counting Regions Formed by Straight Lines [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? Counting School Supplies for the First Days of School [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? Cubes in a Big Cube [03/11/2002] Is there a formula for the number of cubes in an n*n*n cube? Cubes in a Grid [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? Cutting a Square [05/05/1999] What is the maximum number of pieces that a square can be cut into with 70 straight cuts? Declining Balance Interest [03/22/1999] Can you explain declining balance interest to a high school business class? Defining a Sequence [11/04/1998] Can you define sequence, series, convergence, and divergence, and explain how they correlate to one another? Defining Quadratic Formula [01/21/2002] Taking successive differences in a sequence. Deriving any Exponential Formula from a Pattern [01/18/2003] I would like to know the standard process for deriving an exponential formula from an iterative formula. Deriving Trig Functions; Taylor Series [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? Diagonal Sum in Pascal's Triangle [04/02/2001] Find the sum of the reciprocals of the diagonals in Pascal's triangle. Page: []

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