See also the
Dr. Math FAQ:
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.
- Average Yearly Depreciation [06/07/2002]
I'm trying to figure out the average depreciation per year for an
automobile, given its price history over several years.
- 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?
- Card Stacking Problem, Redux [01/31/2017]
A teen seeks clarity on the induction steps that underpin the gravity-defying conundrum of objects balanced over an edge. By switching from torques to centers of mass, Doctor Rick works up an explicit treatment of the recursion often left as an exercise to the reader.
- 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
- 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
- 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
- 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
- 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?