Arithmetic Sequence Calculator

As we all know that the ordered list of numbers is known as a sequence. An arithmetic sequence is one of the types of sequence in math. It is defined as the ordered list of numbers with a certain pattern. In an Arithmetic Sequence, the difference between one term and the next is a constant. The general form of writing an arithmetic sequence is as below:

{a, a+d, a+2d, a+3d, ... }

where,
a is the first term, and
d is the difference between the terms (called the "common difference")

The formula to find the arithmetic sequence is given as,

Formula 1: This arithmetic sequence formula is referred to as the nth term formula of an arithmetic progression.

a_n = a₁ + (n-1)d

where,
a_n = nth term,
a₁ = first term, and
d is the common difference

Formula 2: The formula to find the sum of first n terms in an arithmetic sequence is given as,

S_n = n/2[2a + (n-1)d]

where,
S_n = sum of n terms
a₁ = first term
d is the common difference between the successive terms

Formula 3: The formula for calculating the common difference of an AP (Arithmetic Progression) is given as,

d= a_n - a_n-1

where,
a_n = nth term,
a_n-1 = second last term, and
d is the common difference

Formula 4: The formula for summation of first n terms of an arithmetic progression when the first and the last terms are known is,

S_n = n/2[a₁ + a_n]

where,
S_n = Sum of first n terms
a_n= last term
a₁= first term

It takes much time to find the highest nth term of a sequence. On a general note, it is sufficient if you add the n-1th term common differences to the first term.

In case all the common differences are positive or negative, the formula that is applicable to find the arithmetic sequence is a_n = a₁+(n-1)d. It is also used for calculating the nth term of a sequence. In case of the zero difference, the numbers are equal and there is no need to do further calculations.

• The process to find the summation of an arithmetic sequence is easy and simple if you follow our steps. They are as below
• To solve the summation of a sequence, you need to add the first and last term of the sequence.
• Later, multiply them with the number of pairs. Mathematically, S = n/2 * (a₁ + a)
• If you substitute the value of arithmetic sequence of the nth term, we obtain S = n/2 * [2a₁ + (n-1)d] after simplification. By using this formula, we can easily find the summation of arithmetic sequences.

For practical understanding of the concept, go with our Arithmetic Sequence Calculator and provide the input list of numbers and make your calculations easier at a faster pace. Also, look at the below solved example and learn how to find arithmetic sequences manually. 

Example:

Find the sum of the arithmetic sequence of 2,4,6,8,10,12,14,16? 

Solution:

Given series is 2,4,6,8,10,12,14,16

a is the first term and d is the common difference

The general representation of arithmetic series is a, a + d, a + 2d,...a + d(n−1)

n is the total numbers in that series

As per the rule or formula, we can write an Arithmetic Sequence as: 

x_n = a + d(n−1) (We use "n−1" because d is not used in the 1st term)

(16) = (2) + d(8-1)

7d = (16)-(2)

d = (16)-(2) / 7

d = 2

By using the formula, we can find the summation of the terms of this arithmetic sequence. So, _k=1Σ^n-1 a+kd = n / 2(2a+(n-1)d)
= _k=1Σ⁷ (2)+k × (2) = 8 / 2(2(2)+(8-1)(2)) [put the values in the formula and simplify]
= 8 / 2(2(2)+(7)(2))
= 8 / 2(2(2)+(14))
= 8 / 2((4)+14)
= 8 / 2(18)
= 144 / 2
= 72.0

Hence, the Solution of Arithmetic Sequence of 2,4,6,8,10,12,14,16 is 72.0. 

Example for Finding nth term of Arithmetic Sequence

Example for Finding Sum of n terms of Arithmetic Sequence