# Arithmetic sequence calculator

## Arithmetic sequence

With this calculator find the nth term of arithmetic sequence, calculate common difference, sum of arithmetic sequence.

Contents:

- What is arithmetic sequence?
- How to find the arithmetic sequence?
- What is the common difference in arithmetic sequence?
- How to find common difference in arithmetic sequence?
- What is sum of arithmetic sequence?
- How to find S in arithmetic sequence?

### What is arithmetic sequence?

Arithmetic sequence (algebraic) or arithmetic progression is a list of numbers (progression members) in which each number, starting from the second, is obtained from the previous one by adding to it a constant number d known as **common difference**.

**Definition:**An arithmetic sequence is a sequence of the form a, \ a+d, \ a+2d, \ a+3d, \ a+4d...

If the first term a and common difference d of the arithmetic sequence are known, then it is possible to calculate any member of the arithmetic sequence: a_1 \ a_2 = a_1+d \ a_3 = a_2+d=a_1+2d \ a_4=a_3+d=a_1+3d \ ...

### How to find the arithmetic sequence?

The nth term of the arithmetic sequence can be obtained by adding (n − 1) differences to the first term of the progression.

The general formula of the arithmetic sequence: a_n = a+d*(n-1) where n - is the nth term of an arithmetic sequence, a - is the first member of the sequence, d - is the common difference.

**Example:**

given an arithmetic sequence ( a_n ), where a = 0 and d = 2.

**Find the 10th element of an arithmetic sequence**

### What is the common difference in arithmetic sequence?

The common difference-(d, step, progression difference) is the difference between the next and the previous term of the arithmetic sequence.

If the **common difference** of the arithmetic sequence is positive, then such a sequence called **increasing arithmetic sequence**, if the difference is negative, then **decreasing arithmetic sequence**.

### How to find common difference in arithmetic sequence?

The common difference of the arithmetic sequence can be calculated using the following formulas: d = a_{n+1} - a_n

- d - common difference
- n - nth term of arithmetic sequence

- m - m-th term of arithmetic sequence

- S - sum of arithmetic sequence
- a - first element of arithmetic sequence

### What is sum of arithmetic sequence?

The sum of an arithmetic progression is the result of the addition of all terms in a row. S_n = \displaystyle\sum_{i=1}^{n} a_i = {a_1 + a_n \over 2}n={2a_1 + d(n-1) \over 2}n={a_n -d(n-1) \over 2}n

##### Arithmetic sequence sum formula:

S_n = n* \left(\dfrac{a+a_n}{2}\right) S_n = \dfrac{n}{2}*(2a+d*(n-1))