Arithmetic sequence

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 $a_n$ is defined recursively by a first term $a_0$ and $a_n = a_{n-1} + d$ for $n \geq 1$ , where $d$ is the common difference. Explicitly, it can be defined as $a_n=a_0+dn$ .

Terms in an Arithmetic Sequence

To find the $n^{th}$ term in an arithmetic sequence, you use the formula $a_n = a_1 + d(n-1)$

where  is the  term,  is the first term, and  is the difference between consecutive terms.

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, $s_n=\frac{n}{2}(a_1+a_n)$ . For example,

$5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77$

or

$\frac{7}{2}(5+17)=77$

Example Problems and Solutions

Introductory Problems

Intermediate Problems

Find the roots of the polynomial $x^5-5x^4-35x^3+ax^2+bx+c$ , given that the roots form an arithmetic progression.

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