Q1. Which term of the AP 3, 8, 13, … is 78?
a = 3, d = 5, aₙ = 78
78 = 3 + (n − 1)×5 ⇒ 75 = 5(n − 1) ⇒ n − 1 = 15
n = 16, so 78 is the 16th term.
Arithmetic Progressions — Class 10
An AP just adds the same number each time. Compute any term and the sum instantly, then practise the board numericals.
Arithmetic Progression (AP)A sequence where each term differs from the previous by a fixed number d (the common difference): a, a+d, a+2d, …
nth termaₙ = a + (n − 1)d, where a is the first term.
Sum of n termsSₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
AP calculator Interactive
Enter the first term, common difference and n to get the nth term and the sum.
nth term, aₙ = a+(n−1)d—
Sum, Sₙ = n/2[2a+(n−1)d]—
To check if a number belongs to an AP, solve a + (n−1)d = that number for n — it must be a positive whole number.
Numericals practice Interactive
nth-term and sum problems with steps.
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Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| nth term | aₙ = a + (n − 1)d |
| Sum of n terms | Sₙ = n/2 [2a + (n − 1)d] |
| Sum (with last term l) | Sₙ = n/2 (a + l) |
Common mistakes & exam wins
- First check the common difference d = a₂ − a₁ is the same throughout.
- Use Sₙ = n/2(a + l) when you know the first and last terms.
- The sum of the first n natural numbers is n(n+1)/2 — a special AP.
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 and d the common difference.
What is the sum of n terms of an AP?
Sₙ = n/2 [2a + (n − 1)d], or Sₙ = n/2 (a + l) if the last term l is known.
How do you find the common difference?
Subtract any term from the next one: d = aₙ₊₁ − aₙ.