Difference between revisions of "Arithmetic sequence"
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* [[2006_AMC_10A_Problems/Problem_19 | 2006 AMC 10A Problem 19]] | * [[2006_AMC_10A_Problems/Problem_19 | 2006 AMC 10A Problem 19]] | ||
| + | === Intermediate Problems === | ||
| + | * Find the roots of the polynomial <math>x^5-5x^4-35x^3+ax^2+bx+c</math>, given that the roots form an arithmetic progression. | ||
==See Also== | ==See Also== | ||
Revision as of 19:57, 15 October 2007
Contents
Definition
An arithmetic sequence is a sequence of numbers in which each term is given by adding a fixed value to the previous term. For example, -2, 1, 4, 7, 10, ... is an arithmetic sequence because each term is three more than the previous term. In this case, 3 is called the common difference of the sequence. More formally, an arithmetic sequence 
is defined recursively by a first term 
and 
for 
, where 
is the common difference. Explicitly, it can be defined as 
.
Sums of Arithmetic Sequences
There are many ways of calculating the sum of the terms of a finite arithmetic sequence. Perhaps the simplest is to take the average, or arithmetic mean, of the first and last term and to multiply this by the number of terms. Formally, 
. For example,
or
Example Problems and Solutions
Introductory Problems
Intermediate Problems
- Find the roots of the polynomial
, given that the roots form an arithmetic progression.
, given that the roots form an arithmetic progression.
