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Dr. Math FAQ:
Browse High School Sequences, Series
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Selected answers to common questions:
Strategies for finding sequences.
- Divergent Infinite Series [05/30/2003]
I bought a book _Praise for The Mathematical Universe_, by William
Dunham, with a chapter on Euler's infinite series. A proof he outlined
that I could not follow is this...
- Dividing a Circle using Six Lines [08/29/2001]
What is the largest number of regions into which you can divide a circle
using six lines?
- Does the Series cos(n)/n^(3/4) Converge or Diverge? [11/10/2009]
Doctor Jordan invokes the Euler equation to bound a doozy of a series.
- Do I Use n Or n-1 to Find the nth Term in a Geometric Sequence? [12/01/2009]
A look at how the formula for the nth term in a geometric sequence,
a*r^(n-1), sometimes needs to be a*r^n to fit the problem context.
- Doubling Pennies [11/26/1996]
If I start with a penny and double it daily for 30 days, how many pennies
do I have at the end?
- e as a Series and a Limit [03/30/1998]
Why does e = 1 + 1/2! + 1/3! + 1/4! + ... and lim (1 + 1/n) ^ n, as n --
- Equation of a Sequence with Constant Third Differences [05/26/1998]
Using the method of difference or the Gregory-Newton formula.
- Euler's Faith and Folly [03/25/2011]
What did Euler wrongly assume when he first derived pi^2/6 from the infinite sum of 1/n^2? Doctor Jordan reveals two missteps initially committed by the famous mathematician on this now-classic result.
- Euler's summmation of 1/n^2 [03/15/2000]
Prove that pi^2/6 = the summation of 1/n^2 from 1 to infinity.
- Evaluating Indefinite Sums [12/07/2003]
How can I evaluate the sum of the terms 1/(3n+1)(4n+2), where n ranges
from -infinity to +infinity?
- Evaluating the Series n^2/2^n a Differential Way [11/05/2010]
A student knows that the series n^2/2^n converges as n goes from zero to infinity.
Doctor Ali offers one approach for determining its sum, based on differentiating the
geometric series and its closed form solution.
- Expansion of (x+y)^(1/2) [06/07/1999]
Is there a way to expand (x+y)^(1/2)? If so, how is it derived?
- Expected Tosses for Consecutive Heads with a Fair Coin [06/29/2004]
What is the expected number of times a person must toss a fair coin to
get 2 consecutive heads?
- Exponential Generating Function [05/06/2000]
How can I prove that the exponential generating function of the series 1,
1*3, 1*3*5, 1*3*5*7, ... is 1/sqrt(1-2*x)?
- Exponential Series Proof [05/05/2001]
Given e^x greater than or equal to 1 + x for all real values of x,and
that (1+1)(1+(1/2))(1+(1/3))...(1+(1/n)) = n+1, prove that e^(1+(1/2)+
(1/3)+...+(1/n)) is greater than n. Also, find a value of n for which
1=(1/2)+(1/3)+...+(1/n) is greater than 100.
- Factors and Multiples - Hamiltonian Path [11/02/1998]
We have to make a sequence of numbers, all different, each of which is a
factor or a multiple of the one preceding it.
- Feeding Chickens - Arithmetical Progression [7/6/1996]
A farmer has 3000 hens. Each week he sells 20... what is the total cost
of feeding the hens...?
- A Fibonacci Proof by Induction [06/05/1998]
Let u_1, u_2, ... be the Fibonacci sequence. Prove by induction...
- Fibonacci Sequence - An Example [05/12/1999]
Glass plates and reflections.
- Figurate and Polygonal Numbers [11/21/1998]
I need to know everything about figurate numbers.
- Finding a Formula for a Number Pattern [09/30/2004]
We are learning about sequences and how to find the patterns in
numbers. Our teacher gave us the sequence 0, 3, 8, 15, 24, 35 and
told us that we had to use factoring to find the answer. I know the
answer is (n + 1)(n - 1), but I can't see how to get that.
- Finding a Function to Generate a Particular Output [09/21/2004]
Dr. Vogler presents several possible functions f(n) that will generate
the output 0,0,1,1,0,0,1,1,0,0... for n = 1 to infinity.
- Finding an Explicit Formula for a Recursive Series [05/17/2000]
How far will a man end up from his home if he walks a mile west, then
walks east one half that distance, then walks west half of the distance
he has just walked, and so on?
- Finding a Non-Recursive Formula [06/10/1999]
How can I find a non-recursive formula for the recurrence relation s_n =
- [s_(n-1)] - n^2 with the initial condition s_0 = 3?
- Finding an Unknown Sequence [3/31/1996]
I can't figure out where to start with this Series and Sequences
question: 1+3x+6(x)(x)+10(x)(x)(x)+15(x)(x)(x)(x)+. . .
- Finding a Series Given the Sum [09/27/1999]
How can I find all series of consecutive integers whose sum is a given
- Finding a Term of an Arithmetic Series [12/13/1995]
The fifth term of an arithmetic series is 16 and the sum of the first 10
terms is 145. Write the first three terms.
- Finding Catalan Numbers [12/15/1999]
What are Catalan numbers and what applications do we have for them?
- Finding Common Numbers in Two Sequences [09/21/2006]
I'm working with sequences that start with an initial value and an
initial amount to add to get the next term. The amount added then
increases by 2 as you move from term to term. If I have two such
sequences, is there a way to calculate what numbers they will have in
common based on the two initial values and amounts to add?
- Finding Number Patterns [05/29/1999]
I am trying to find the pattern of the numbers
- Finding Rules for Number Patterns [06/05/2009]
I'm having trouble finding an algebraic expression that generates the
pattern 3, 5, 8, 12, 17, 23, 30. Can you help?
- Finding Sums of Sines and Series [03/10/2004]
I am trying to find the sum of sin1 + sin2 + sin3 + ... + sin90. I'm
also trying to find the sum of 1^n + 2^n + 3^n + 4^n + ... + n^n. Can
you help me?
- Finding the 1000th Term in a Sequence [1/19/1996]
Two kids on a car trip decide to count telephone poles. One kid counts
normally, 1,2,3,4,5...25,26,27...31,32,33, etc. The other kid counts them
a different way: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2,
- Finding the Digit of a Decimal Expansion [11/14/1998]
What digit will appear in the 534th place after the decimal point in the
decimal representation of 5/13?
- Finding the Missing Numbers in a Sequence [11/30/1995]
Fill in the blanks for this series of numbers based on its underlying
pattern: 3, 4, 6, 8, 12, (), 18, 20, (), 30, 32
- Finding the Next Number in a Sequence Given Its Geometric Mean ... Which Is a Square Root [09/24/2009]
A student who knows how to calculate geometric means gets rattled when
trying to determine a sequence from its square root geometric mean.
- Finding the Next Number in a Series [07/22/2002]
Are there any formal or systematic methods for solving problems that
ask you to find the next number in a series?
- Finding the Rule for a Given Sequence [09/11/2008]
If the first six terms of a sequence are -4, 0, 6, 14, 24, 36, what is
the rule? Find the 20th and 200th terms. This answer discusses finite
differences and other handy techniques for solving this sort of problem.
- 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.