Arithmetic Sequence Calculator

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Need to find the nth term or sum of an arithmetic sequence? This arithmetic sequence calculator computes any term and the sum of the first n terms instantly.

Arithmetic Sequence Calculator

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Using the calculator

Enter three values:

First term (a₁) – the starting value of your sequence
Common difference (d) – the constant amount added to each term
Number of terms (n) – how many terms to consider

The calculator instantly shows the nth term (aₙ), the sum of all n terms (Sₙ), and optionally the full sequence.

What is an arithmetic sequence?

An arithmetic sequence (or arithmetic progression) is a sequence where each term differs from the previous one by a constant amount called the common difference.

Examples:

2, 5, 8, 11, 14, ... (first term = 2, common difference = 3)
10, 7, 4, 1, -2, ... (first term = 10, common difference = -3)
0, 0.5, 1, 1.5, 2, ... (first term = 0, common difference = 0.5)

The nth term formula

To find any term in an arithmetic sequence without listing them all:

a_n = a_1 + (n - 1) \cdot d

Where:

aₙ = the nth term
a₁ = first term
d = common difference
n = term position

Example: finding the 50th term

For the sequence 3, 7, 11, 15, ... (a₁ = 3, d = 4), find the 50th term:

a_{50} = 3 + (50 - 1) \times 4 = 3 + 196 = 199

The sum formula

To find the sum of the first n terms:

S_n = \frac{n(a_1 + a_n)}{2}

This is often written as:

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

The intuition: pair the first and last terms (they sum to the same value as the second and second-to-last, etc.), then multiply by the number of pairs.

Example: sum of first 100 positive integers

The sequence 1, 2, 3, ..., 100 is arithmetic with a₁ = 1 and d = 1:

S_{100} = \frac{100(1 + 100)}{2} = \frac{100 \times 101}{2} = 5050

This is the famous result attributed to young Gauss, who supposedly computed it in seconds by recognizing the pattern.

Common applications

Arithmetic sequences appear in many real-world contexts:

Linear depreciation: An asset loses the same dollar amount each year
Salary increases: Fixed annual raises (e.g., $2,000/year)
Seating arrangements: Stadium rows with increasing seats per row
Loan payments: Simple interest with equal principal payments

Arithmetic vs geometric sequences

Don't confuse arithmetic and geometric sequences:

Arithmetic: Add the same amount each time (2, 5, 8, 11...)
Geometric: Multiply by the same ratio each time (2, 6, 18, 54...)

Arithmetic sequences grow linearly; geometric sequences grow exponentially.

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PKPK started DQYDJ in 2009 to research and discuss finance and investing and help answer financial questions. He's expanded DQYDJ to build visualizations, calculators, and interactive tools. 
 
 PK lives in New Hampshire with his wife, kids, and dog.

Arithmetic Sequence Calculator