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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.

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

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

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

Newton's Method and Continued Fractions [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)?

Nonhomogeneous Linear Recurrence Relations [05/18/2004]
Given a recursive formula: a(n+1) = a(n) + (a(n) - b)*t, where b is a known constant and a(1) is also known, I am trying to find the explicit formula like y = ????? * t^n.

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

Nth Term in a Sequence [01/11/2003]
Here is the sequence: 1, 2, 5, 14 ... Find the following 2 terms and a formula for the nth term.

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

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

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

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

Pages in a Book [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?

Paper-folding [12/30/1994]
My students and I have done the paper-folding thing several times, and we thought it was only possible to fold a single sheet of paper 7 or 8 times. A couple of days ago on TV they said 10 times! Do you agree?

Pattern of Remainders [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?

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

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

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

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

Prize Money [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?

Probability of Random Numbers Being Coprime [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?

Program to Calculate Pi [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 by Induction [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?

Proof by Induction [03/16/2003]
Evaluating a summation: 2^r as r goes from 0 to n.

Proof Involving Sums of Reciprocals [09/13/2004]
For n > 1, prove that the sum of reciprocals from 1 to n does not sum to an integer. For n = 4, for example, prove that 1/1 + 1/2 + 1/3 + 1/4 does not sum to an integer.

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

Proof of Series ln(1+x) [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.

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

Proof of Stirling's Approximation [03/09/2006]
Can you prove that lim ((e^n)(n!)) / ((n^n)(n)^1/2 = (2pi)^1/2 ?

Proof of the Infinite Series That Calculates 'e' [02/04/2004]
Is there a proof about this infinite series that gives the value of e: 1 + 1/1! + 1/2! + 1/3! + 1/4! + . . . + 1/n! where n goes to infinity?

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

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

Proof Using Pell's Equation [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?

Proving Series Convergence [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.

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

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

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

Rat Population [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?

Reasoning Out a Number Pattern Formula [06/02/2005]
I need to find a formula for finding next term and the nth term in the sequence 1,5,9,13,17,21.

Reciprocals of Integers Greater Than 1 as Sum of a Series [07/01/2004]
Show that the reciprocal of every integer greater than 1 is the sum of a finite number of consecutive terms of the series 1/[j(j + 1)].

Recursion, and Closure [11/04/2015]
Having sussed out a recursive formula, a teen wonders how to determine its equivalent closed form — and seeks a more methodical approach to such problems. After a few observations about finite differences, Doctor Greenie identifies a template for closed form formulas, and proceeds to test it.

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