How do you find the common difference of an AP?

d = (last term − first term) / (n − 1), where n is the total number of terms. Example: for the AP 5, 8, 11, 14 (4 terms): d = (14 − 5) / (4 − 1) = 9/3 = 3.

Arithmetic Progression Calculator – nth Term, Sum, AP Formula

Q: What is the formula for the sum of n terms of an AP?

Sₙ = n/2 × (2a + (n−1)d), or equivalently Sₙ = n/2 × (first term + last term). Example: sum of first 10 terms of 3,7,11,15…: S₁₀ = 10/2 × (6 + 36) = 5 × 42 = 210.

Q: How do you find which term a value is in an AP?

Use n = (t − a)/d + 1, where t is the term value, a is the first term, and d is the common difference. If n is a positive integer, t is in the AP; otherwise it is not.

Q: What is the difference between AP and GP?

In an Arithmetic Progression (AP), consecutive terms differ by a constant value called the common difference (d). In a Geometric Progression (GP), consecutive terms have a constant ratio called the common ratio (r). Example AP: 2, 5, 8, 11 (d=3). Example GP: 2, 6, 18, 54 (r=3).

Enter the first term (a), common difference (d), and term number (n) to find aₙ and Sₙ.

First term (a)

Common difference (d)

Term number (n)

Please enter valid numbers. n must be a positive integer.

Given a term value, find which position it occupies in the AP.

First term (a)

Common difference (d)

Term value (t)

Please enter valid numbers. d cannot be 0 (use Simple Probability tab instead).

Given the first term, last term, and number of terms, find d and the sum Sₙ.

First term (a)

Last term (l)

Number of terms (n)

Please enter valid numbers. n must be an integer ≥ 2.

How to Use

Choose a Mode

Pick nth Term & Sum for standard AP questions, Find Term Number to find which term equals a value, or Find Common Difference when first and last terms are known.

Enter the Known Values

Fill in the first term (a), common difference (d), and term number (n) — or whichever three values your problem gives you.

Read the Solution

Get the nth term (aₙ), sum (Sₙ), and a complete step-by-step breakdown with the AP series listed.

Frequently Asked Questions

What is the formula for the nth term of an AP?

aₙ = a + (n − 1) × d, where a is the first term, d is the common difference, and n is the term position. Example: AP 3, 7, 11, 15… → a = 3, d = 4, so the 10th term = 3 + 9 × 4 = 39.

What is the formula for the sum of n terms of an AP?

Sₙ = n/2 × (2a + (n − 1)d). Equivalently, Sₙ = n/2 × (first term + last term). Example: sum of first 10 terms of 3, 7, 11… → S₁₀ = 10/2 × (6 + 36) = 5 × 42 = 210.

How do you find the common difference?

d = (last term − first term) / (n − 1). Example: AP 5, 8, 11, 14 has 4 terms → d = (14 − 5) / (4 − 1) = 9/3 = 3. You can verify: each consecutive pair differs by 3.

How do you find which term a value is in an AP?

Set aₙ = t and solve: n = (t − a)/d + 1. If the result is a positive integer, t belongs to the AP at position n. If not a positive integer, t is not a term of the AP. Example: is 39 a term of 3, 7, 11…? n = (39 − 3)/4 + 1 = 36/4 + 1 = 10. Yes, 39 is the 10th term.

What is the difference between AP and GP?

In an AP, terms differ by a constant (common difference d): 2, 5, 8, 11 (d = 3). In a GP, terms multiply by a constant (common ratio r): 2, 6, 18, 54 (r = 3). The AP sum formula is Sₙ = n/2 × (2a + (n−1)d); the GP sum formula is Sₙ = a(rⁿ − 1)/(r − 1).

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