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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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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 checkerboard?

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 some series?

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 generation n?

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 class?

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?

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