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Dr. Math FAQ:
Browse High School Sequences, Series
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Selected answers to common questions:
Strategies for finding sequences.
- 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
- 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
- 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
- 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
- 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.
- Differences Method [07/06/2001]
How does the method of differences work?