Arithmetic sequence formulas and examples

A sequence of numbers such that the difference between the consecutive terms is a constant is called arithmetic progression or arithmetic sequence.

For example, the sequence 2, 5, 8, 11, 14, 17, 20... is an arithmetic progression with a common difference of 3.

Rule of arithmetic sequence

General rule of arithmetic sequence can be written by this formula, where $a_1$ - the first term, $d$ - the difference between the terms, $n$ - the number of term.

$$a_n=a_1+(n-1)d$$

Write a rule, and calculate the 40th term, for this arithmetic sequence: 3, 10, 17, 24, 31...

The first term value of this sequence is $a_1=3$. This sequence has a difference of 7 between each number, so $d=7$. Using the arithmetic sequence rule we get:

$$a_n=a_1+(n-1)d=3+(n-1)7=7n-4$$

The 40th term of this arithmetic sequence can be find easily using expression above:

$$a_{40}=7\cdot 40-4=276$$

Let we have arithmetic sequence where $a_1=-102$ and $d=8$. What is the number of term with value 282?

Using the arithmetic sequence rule we can write simple equation and solve it

\begin{align} a_n & =a_1+(n-1)d \\ 282 & =-102+(n-1)8 \\ n & =49 \end{align}

Sum of arithmetic sequence

To sum up the terms of an arithmetic sequence is simple only when we have not many terms. But for sequences that have large number of terms we can use this formula below.

$$S_n=\frac{2a_1+(n-1)d}{2}n$$

Let we have arithmetic sequence where $a_1=5$ and $d=3$. What is the sum of 100 first terms in this sequence?

Using the arithmetic sequence sum formula we get:

$$S_{100}=\frac{2\cdot 5+(100-1)3}{2}\cdot 100=15350$$

2021-07-29