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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.
 NonRecursive Formula [06/05/2001]

I want to know the nonrecursive 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.
 OnetoOne 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 11 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?
 Paperfolding [12/30/1994]

My students and I have done the paperfolding 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 n1 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/(1x)^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 TI83 calculator to calculate the
infinite digits of Pi while displaying them onscreen.
 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 xx^2/2+x^3/3x^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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