Arithmetic Sequence Calculator

Arithmetic Sequence Calculator

Enter the first number, common difference, and the nth number in the arithmetic series calculator and find the nth term of an arithmetic sequence.

Choose Calculation Mode

Common difference:

The first number (d):

Number of terms (n):

Choose what to compute:

Find \(a_n\) Find \(S_n\)

Enter nᵗʰ Term:

a( ) =

Sum of First n Terms:

S( ) =

First Term (a₁):

Common Difference (d):

Sum of first n terms

Arithmetic Sequence Calculator

The Arithmetic Sequence Calculator helps you find the nth term and the sum of the first n terms of an arithmetic sequence. By entering the first term, common difference (d), and the number of terms, you can instantly compute accurate results along with step-by-step solutions.

What Is an Arithmetic Sequence?

In mathematics:

An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a constant value (called the common difference) to the previous term.

Arithmetic Sequence

The constant value is denoted by d. The sequence can increase (if d is positive) or decrease (if d is negative). Arithmetic sequence, arithmetic series, and arithmetic progression are closely related terms.

Arithmetic Sequence Formulas

1. Formula for the nth Term

a_n = a₁ + (n − 1)d

2. Formula for the Sum of the First n Terms

S_n = (n / 2) [2a₁ + (n − 1)d]

Where:

a_n = nth term
a₁ = first term
d = common difference
n = number of terms
S_n = sum of first n terms

How To Calculate the nth Term

Example 1

Find the 9th term of the sequence: 3, 8, 13, 18, 23, ...

Step 1: Identify values

a₁ = 3
d = 5
n = 9

Step 2: Apply the formula

a₉ = 3 + (9 − 1) × 5

a₉ = 3 + 8 × 5

a₉ = 3 + 40

a₉ = 43

Final Answer: The 9th term is 43.

How To Calculate the Sum of First n Terms

Example 2

Find the sum of the first 10 terms of the sequence: 1, 4, 7, 10, ...

Step 1: Identify values

a₁ = 1
d = 3
n = 10

Step 2: Apply the formula

S₁₀ = (10 / 2) [2(1) + (10 − 1) × 3]

S₁₀ = 5 [2 + 27]

S₁₀ = 5 × 29

S₁₀ = 145

Final Answer: The sum of the first 10 terms is 145.

Arithmetic Series to Infinity

An arithmetic sequence does not have a finite sum to infinity unless the common difference d = 0. If d ≠ 0, the terms continue increasing or decreasing indefinitely, and the series diverges.

Example Table of an Arithmetic Sequence

Consider a sequence where a₁ = 5 and d = 4:

n	a_n
1	5
2	9
3	13
4	17
5	21
10	41
20	81
30	121
40	161

Why Use an Arithmetic Sequence Calculator?

Instantly calculates nth term
Finds the sum of n terms accurately
Reduces calculation errors
Provides step-by-step solutions

Simply enter the first term, common difference, and number of terms to compute results quickly and efficiently.

Arithmetic Sequence Calculator

Arithmetic Sequence Calculator

What Is an Arithmetic Sequence?

Arithmetic Sequence Formulas

1. Formula for the nth Term

2. Formula for the Sum of the First n Terms

How To Calculate the nth Term

Example 1

How To Calculate the Sum of First n Terms

Example 2

Arithmetic Series to Infinity

Example Table of an Arithmetic Sequence

Why Use an Arithmetic Sequence Calculator?

References