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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.
- Adding Arithmetic Sequences [07/10/1998]
How do you add the numbers from 1 to 5000 without actually doing it or
using a calculator? What if you were adding just the odd numbers?
- Calculating the Fibonacci Sequence [11/28/1996]
Is there a formula to calculate the nth Fibonacci number?
- Decimal To Fraction Conversion [06/25/1998]
I am trying to find a method (one that can be programmed on a PC) to
convert the decimal part of a real number to a fraction represented by
integers for the numerator and denominator.
- Describing Patterns in Sequences [04/16/2002]
My students are able to identify the number patterns corresponding
to number sequences, but are having difficulty explaining them in
- Doubling Grains of Wheat [10/7/1996]
A man asked for 1 grain of wheat for the 1st square on a chess board, 2
grains for the 2nd square...
- Doubling Sequence [8/24/1996]
On Jan 1st it snowed one centimeter; on Jan 2, 2cm; on Jan 3, 4 cm...
- Fibonacci and Incoming Bits [09/08/99]
Given a transmitter sending 100 bits of random data over an ideal
communication channel, what is the probability that there will be three
consecutive 1's at least once in the sequence?
- Finding a Pattern [11/11/2001]
Give the next four numbers in the sequence: 2, 8, 7, 28.
- Finding a Rule for a Sequence [07/24/2003]
What is the next number in this sequence? 1, 3, 11, 67, ?
- Finding Sum Formula using Sequences of Differences [06/28/1998]
Finding a formula for the sum of the first n fourth powers using
sequences of differences.
- Finding the Pattern in a Series of Numbers [11/14/1995]
What is the pattern for 1, 8, 27...?
- Infinite square root [6/4/1996]
If y= sqrt(2+ sqrt(2+ sqrt(2+ sqrt(2+ ..., y=2,... how can I prove that
this is true, using normal properties of roots?
- Look-and-Say Sequence [02/14/2002]
I can't find the next six numbers: 1, 11, 21, 1211, 111221, ...
- Mean Proportionals and Geometric Means [01/06/1999]
How do you find the mean proportional of two numbers? What about two mean
proportionals? n mean proportionals?
- Next Number in a Sequence [03/13/2002]
Given any sequence, one can construct an infinite number of n-degree
polynomials that satisfy the sequence, hence discern an infinite number
of answers. What is the proof for this?
- Sequence Differences [06/24/2003]
The third and fourth terms of a sequence are 26 and 40. If the second
differences are a constant 4, what are the first five terms of the
- Strategies for Tests on Sequences [7/9/1996]
I have a problem answering test questions about number sequences.
- Sum of n Odd Numbers [7/11/1996]
Why is the sum of the first n odd numbers the square of n?
- The Traveling Bee [09/18/1998]
If a bee travels between two trains that are moving at 30 and 20 mi/hr
respectively, starting from 50 mi apart, how far does the bee travel?
- Unsolvable Equations [11/10/2001]
If I have an equation in the form of x^n+y^n=z, how do I solve for n?
- Why is Zero the Limit? [02/25/2002]
Why is zero called the limit of the terms in the sequence the limit of 1
over n, as n approaches infinity, equals zero?
- 121, 111211, 311221 Puzzle (Look and Say Sequence) [10/23/2001]
121, 111211, 311221 - what's the next number?
- 1 + 2 + 3 + 4 + ... Equals ... -1/12?! [09/18/2012]
Doctor Vogler explains how functions extended by analytic continuation can be
evaluated to produce counterintuitive results.
- 1, 7, 23, 55, 109, 191, ___ [10/03/2002]
My family is stumped on this number pattern: 1,7,23,55,109,191,___ ...
- 21^100 - Last Two Digits [09/04/1997]
What are the last two digits of 21 to the 100th power?
- 22/7 as an Approximation for Pi [04/01/1998]
Approximating pi by simple continued fractions.
- Activities to Find Pi [10/07/1998]
Can you suggest any classroom activities to find pi, other than the
standard way of measuring the circumferences and diameters of circles?
- Advanced Algebra [09/23/1997]
My teacher gave us this problem: 1+1/(1+1/(1+1/(1+1/1+...)))
- Alternating Harmonic Series [11/18/1997]
I am trying to find the proof for the sum of the alternating harmonic
series. I did find out that it is ln(2), but please tell me why?
- Alternating Sequence [01/27/1997]
Find a pattern and the next three numbers in the sequence: 0, 8, 27...
- Ant Walking in a Squared Spiral [06/02/1999]
An ant walks out a distance of 1 from the origin, down the x-axis. It
then turns left and goes up 1/2. If it continues turning left and going
the half the previous distance, where does the ant end up?
- Are All Infinitely Long Repeating Numbers Even? [06/06/2000]
Given an infinitely long repeating series, x = 12341234..., then 10000x =
123412341234... Since 9999 is odd and 12340000... is even, can we say
that x is even, and therefore all infinitely long repeating series are
- Arithmetical Progression [7/7/1996]
An arithmetical progression has a common difference of 1/1/2. The sum of
the first n terms is 365 and the sum of the first 2n terms is 1330.
Calculate the value of n and the first term.
- Arithmetic and Geometric Progressions [03/23/1998]
Given a set of conditions, can you find a specific term in an arithmetic
or geometric progression?
- Arithmetico-Geometric Series and Polylogarithms [07/06/2006]
Is there a closed form expression for the sum of the series
e^(-x) + 1/9 * e^(-3x) + 1/25 * e^(-5x) + 1/49 * e^(-7x) + ... ?
- Arithmetic Progression [12/19/1996]
If (b+c-a)/a, (c+a-b)/b and (a+b-c)/c are in arithmetic progression, show
that 1/a, 1/b and 1/c are also in arithmetic progression.
- Arithmetic Sequence Conundrum [10/11/2002]
For some real number T, the first three terms of an arithmetic
sequence are 2T, 5T - 1, and 6T + 2. What is the numerical value of
the fourth term?
- Arithmetic Sequences as Lines [09/05/2003]
In a sequence like -40, -25, -10, 5, ... is there a sure-fire way to
find the the general term?
- Arithmetic Series [5/19/1996]
How do you calculate a series like 2,4,6,8... for say 3 terms starting
anywhere in the series not by adding 3 specific terms together, but by
using the first term and the number 3?
- Arithmetic vs. Exponential Increases [05/06/1999]
What does "....the work produced... will increase exponentially rather
than arithmetically" mean?