# 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 $a_n$ is the $n^{th}$ term, $a_1$ is the first term, and $d$ is the difference between consecutive terms.

## Sums of Arithmetic Sequences

Main article: Arithmetic series

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

### 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.