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

a_n = a + d(n-1) = \implies a_{10} = 0 + 2 * (10 -1) = 2*9 = 18

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

d = \dfrac{a_n - a_m}{n - m}

m - m-th term of arithmetic sequence

d = \dfrac{2*\dfrac{S_n}{n}-2a}{n-1}

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))

Arithmetic sequence calculator