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High School Archive

Browse High School Sequences, Series
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Selected answers to common questions:
    Strategies for finding sequences.

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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.

Recursive and Explicit Formulas [01/19/1999]

Is there an easy way to convert recursive formulas into explicit ones and vice versa?

Recursive vs. Explicit Formulas [01/02/1997]

What is the difference between explicit and recursive formulas?

Repeating Decimals [04/28/1999]

I am interested in finding longer repeating groups in number tails of repeating decimals.

Repeating Decimals as Geometric Series [02/13/2001]

How can you express repeating decimals as geometric series and convert them to fractions using the series sum formula?

Rule for a Sequence [8/25/1996]

What is the next number of these two sequences? {8 13 5 15 20 12} and {20 25 17 51 56 48}

Scoring System Problem [10/28/2001]

What is the highest score that is impossible to make?

Second-Order Linear Recurrences [06/08/2001]

Three problems involving recurrence equations.

Second Order Recurrence with Non-Constant Coefficients [05/27/2005]

I'm trying to find a closed form solution of a second order recurrence relation with no constant coefficients, specifically: u(n+2) = 2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help?

Sequence and Series Terminology and Concepts [11/27/2005]

I'm studying sequences and series and am confused about how they are defined in terms of functions as there seem to be some inconsistencies. Can you help clarify things for me?

Sequence Missing a Middle Term [02/26/2016]

Finding the last term of a sequence is easy. But what if the unknown term is missing from the middle of the sequence? Doctor Greenie outlines two approaches — one complex, one simple, both robust — emphasizing throughout that there are always alternative solutions that satisfy these kinds of problems.

Sequence of Squares [07/25/1998]

Do you have any information on the sequence of squares?

Sequence of Triangular Numbers [7/13/1995]

What is the sequence called 1, 3, 6, 10, 15 and how is it generated?

Sequence Pattern and Closed Form [7/16/1996]

Given the pattern for a sequence, I can't figure out a general rule for the nth term.

Sequences [7/29/1996]

The sum of three numbers is 147 and when multiplied together they yield 21952... Find a formula for 60, 30, 20, 15...

The Sequence Sin(n) [02/20/2002]

I am trying to prove that the sequence sin(n), for n, a natural number, does not converge.

Sequence Uncertainty [04/03/2013]

A student senses that he has the wrong number to continue a sequence that contains several negative numbers. Doctor Peterson empathizes before picking up on the patterns.

Series Convergence [01/27/2001]

Test these series for convergence; if the series is alternating, tell whether the convergence is conditional or absolute...

Series Convergence [02/28/2001]

Why does 1 + 1/2^z + 1/3^z + ... converge for Re(z) greater than 1?

Series Divergence [03/03/1999]

Show that the series sum(k=0 -> infinity): (k/e)^k/k! is divergent.

Series Expansion of 1/(1-x) [08/01/1998]

Can you explain the series expansion identity 1/(1-x) = 1 + x + x^2 + x^3 + ... ? In what region does it converge?

Series for which Convergence is Unknown [11/09/2000]

Are there series for which it is unknown whether they converge or diverge?

Series Problem: Find the Sum [6/24/1996]

Find sum[sin(nx)/(3^n),(n,0,oo)] if sin x=1/3 and x is in the first quadrant.

A Series that Converges and Diverges? [04/30/2002]

Let N = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + ... Does this series both converge and diverge?

Series Types [05/11/1997]

What are the definitions of convergent, divergent, and oscillating series?

Showing That the Sum of the Infinite Series cos(n)/n Converges [04/01/2008]

I'm trying to determine if the sum of the series (cos n)/n for n = 1 to infinity converges. I've tried some tests but they have been inconclusive.

Sigma Notation [4/14/1996]

I am trying to find questions regarding sigma notation.

Sigma Notation [09/07/2001]

To prove that sigma (i^2) from i = 1 to n i equal to (n(n+1)(2n+1))/6 start with (i+1)^3 - i^3...

A Simple Expression? [1/26/1996]

Is it possible to have a simple expression for a certain series starting at k=b...?

Simple Number Pair Series Yields Surprising Ratio ... Why? [12/31/2009]

An enthusiast wonders about the curious ratio that emerges from a simple pattern for generating number pairs. Doctor Rick builds an algebraic argument for why its phi-like recursive relationship approaches the square root of 2.

Simplify a Geometric Series [05/06/2003]

x^n + x^(n - 1) + x^(n - 2) + ... + x^(n - n)

Solutions to X^Y = Y^X [12/21/2000]

How can I find the solutions to the equation x^y = y^x? I have been told that it involves the Lambert W Relation.

Solving an Equation with Infinite Exponents [03/15/2007]

If x^x^x^x^x^x^x^x^x^x^x...... = 2, solve for x. How can I solve that equation?

Solving a Sequence [12/7/1995]

Write an expression to find the nth term of the following sequence: 3, 9, 18, 30, 45 . . .

Solving Continued Fractions [08/08/1998]

How do you get sqrt(2) from 1/(2 + 1/(2 + 1/(2 + ...)))? How do you solve continuous fractions in general?

Some Algebra Problems [6/1/1996]

If z=(3-2i)^1/2 then find z^-...

Square Root of 3 minus 1 [09/24/1997]

Express sqrt3-1 as a continued fraction.

Square Root Theory [11/16/2001]

When I enter any positive number in the calculator or a fraction like 0.1, then take the square root of that number, then take the square root of that number, and keep pressing the square root button over and over, I eventually get to number 1. Why?

Stair Patterns [02/27/2001]

The 1st step is made with 4 matches, the 2nd with 10 matches, the 3rd with 18, the fourth with 28. How many matches would be needed to build 6, 10, and 50 steps?

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