Sum of first n terms of an AP
- CBSE · NCERT Class 10, Ch 5 Arithmetic Progressions
The sum of the first n terms of an AP is Sₙ = n/2 [2a + (n - 1)d].
Here a is the first term and d the common difference. When the last term l is known, a shorter form works: Sₙ = n/2 (a + l). Both come from one idea. Pair the first term with the last, and the second with the second-last. Every pair has the same total.
The chapter tells how Gauss, aged 10, added 1 to 100 this way. There are 50 pairs, each adding to 101, so the total is 5050.
A small example
Add 20 + 40 + 60 + ... + 380. There are 19 terms, with a = 20 and l = 380. So S₁₉ = 19/2 × (20 + 380) = 19/2 × 400 = 3800. The nth term formula is what tells you there are 19 terms.
Where marks go
That sum is Question 28(B) of CBSE's 2026-27 Maths Standard sample paper. After n = 19 is found, the board's Maths Standard marking scheme for 2026-27 gives half a mark for the formula with the numbers in. The final half mark is for 3800.
In Question 37 of the 2026-27 Maths Basic sample paper, part (iii)(A) asks for the total trees planted by 12 classes. The marking scheme gives 1 mark for S₁₂ = 12/2 {2(10) + (12 - 1)4}, and 1 mark for 384. In both papers, the formula with the right values is a mark in itself. Our guide shows how step marking splits an answer.
Sources
Facts last checked against these sources on 30 September 2026.
About the author
Chief Architect
An engineer (B.Tech, Computer Science) with over 20 years of preparing students for entrance exams, including as Dean at FIITJEE and Vice-President at Aakash. He is the chief architect of the TRUpreBoards evaluation engine.
Mohit Sardana on LinkedIn
