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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.

Finding the Sum of an Infinite Series [03/05/2006]
Find the sum of the series 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ... which are the reciprocals of the positive integers whose only prime factors are 2's and 3's.

Finding the Sum of Arithmetico-Geometric Series [09/13/2004]
Find the sum of the infinite series 1/7 + 4/(7^2) + 9/(7^3) + 16/(7^4) + ... I would also like to know if there is a general rule to find the sum of (n^2/p^n) for n = 1 to infinity.

Finding the Sum of Arithmetic Series [06/12/2006]
Find the sum of the arithmetic series 4 + 10 + 16 + ... + 58.

Find Sum of First Five Terms [03/26/2003]
The 452nd term of an arithmetic sequence is -893 and the 84th is 27. Find the sum of the first five terms of the sequence.

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

Find the 276th Letter [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,...

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

Finite Differences and Polynomials [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?

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

Formula for Sum of First N squares [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?

Formula for the Nth Term in a Geometric Sequence [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?

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

Formula to Sum a Series of Square Roots [07/03/2004]
Is there a way to calculate the sum of the series sqrt(x) as x goes from 1 to n?

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

Fourier Sine and Cosine Series [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?

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

Fresnel Integral of sin(x^2) [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.

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

From Reduction to Induction [02/03/2011]
Replace any two numbers x and y from (1, 2, ..., n) with the new single quantity x + y + xy; continue in this way until only one number remains. To find a formula for the smallest number possible from this procedure, Doctor Jacques lays the groundwork for a proof by induction.

From Zero to One, Indeterminately [08/13/2015]
A teen struggles to spot the fallacy in a proof that concludes that zero equals one. With examples galore, Doctor Floor reveals the hazards of computing with infinity.

Gauss and the Sum of Numbers from 1 to 100 [03/02/2006]
Can you help me understand how Gauss did his number addition trick in a way that I can explain it to young students?

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

General Approach for Sum of Arithmetic Series [08/24/2006]
What is the formula for figuring out the sum of adding consecutive even numbers? 2 + 4 = 6, 2 + 4 + 6 = 12, 2 + 4 + 6 + 8 = 20, etc.

Generalised 'Fibonacci' Series and Phi [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?

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

Geometric Sequences and Series [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?

Geometric Series and Sequences [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?...

Geometric Series for Catching Fish Each Day [05/24/2007]
Each day, a fisherman catches 40% of the amount of fish he caught the day before. If he catches 31,416 pounds the first day and continues for 20 years, how many total pounds of fish will he catch?

Grains of Wheat [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?

Halving and Halving Again - Zeno's Paradox [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?

Harmonic Series [04/28/2003]
Prove that if in the sum 1 + 1/2 + 1/3 + ... + 1/n we throw out each term that contains 9 as a digit in its denominator, then the sum of the remaining terms is less than 80.

Help Proving that the Sequence for the Natural Log Is Bounded [09/15/2009]
Why do the inequalities 1 - (1/n) < 1, 1 - (2/n) < 1, and so on help me prove that [1 + (1/n)]^n is less than 2 + (1/2!) + (1/3!) + ...... + (1/ n!) ?

House of Cards [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?

How Many Hidden Faces? [07/07/2002]
When you place a number of cubes in a row on a surface, how many of the faces can't be seen from any position? Is there a formula for this?

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

A Hundred-Row Number Pyramid [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?

Increasing and Decreasing Subsequences; Pigeonhole Principle [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?

Infinite Continued Fraction [05/15/2002]
What can you determine about the value of the infinite continued fraction [1;1,2,3,1,2,3,1,2,3....]?

Infinite Product [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 Sequence [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.

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