Class 11 · Mathematics

Sequences and Series (Class 11): AP, GP and the Sums Worth Knowing

A short chapter with a high mark-to-effort ratio — provided the formulas are used with the right n, which is where most of the lost marks actually go.

Class 11 · Mathematics · Boards + JEE · Published 27 August 2026

In short: An AP has a constant difference and a GP a constant ratio. Nearly every question is one of four things: find a term, find a sum, insert means between two numbers, or use AM ≥ GM. The formulas are few; the errors come from off-by-one mistakes in n and from forgetting the |r| < 1 condition on an infinite GP.

The formulas, and the condition each one needs

Quantity	Formula	Condition or trap
n^th term of an AP	a_n = a + (n − 1)d	It is (n − 1), not n
Sum of n terms of an AP	S_n = n/2 [2a + (n − 1)d]	Or n/2 (a + l) when the last term is known
n^th term of a GP	a_n = arⁿ⁻¹	Again (n − 1)
Sum of n terms of a GP	S_n = a(rⁿ − 1)/(r − 1)	Undefined at r = 1; use S_n = na
Sum of an infinite GP	S_∞ = a/(1 − r)	Valid only when \|r\| < 1
AM ≥ GM (two positives)	(a + b)/2 ≥ √(ab)	Equality only when a = b

Where the marks actually go

Three places, and none of them is the formula itself. First, the off-by-one error: “the 10th term” uses n = 10 and therefore (n − 1) = 9, and a surprising number of otherwise correct solutions use 10. Second, the infinite GP condition: writing a/(1 − r) when |r| ≥ 1 gives a finite answer for a series that diverges, and examiners mark it wrong even though the arithmetic is clean. Third, arithmetic slips in the sum formula, which are best caught by checking the answer against a rough estimate rather than by re-doing the calculation the same way.

A trick worth having: when three numbers are in AP, write them as a − d, a, a + d. When three are in GP, write them as a/r, a, ar. The symmetry cancels a variable immediately and turns most “find the numbers” questions into one equation instead of three.

Special sums to have at hand

Σn = n(n + 1)/2
Σn² = n(n + 1)(2n + 1)/6
Σn³ = [n(n + 1)/2]², which is the square of the first sum

These turn up in questions that do not look like sequence questions at all — particularly when a series is given term by term and you are asked for the sum to n terms. Recognising the pattern is most of the work.

AM ≥ GM, and when it is the point of the question

Any question asking for a minimum or maximum of a sum or product of positive quantities is very often an AM–GM question in disguise. If a question gives a fixed product and asks for the least sum, apply the inequality and remember that equality holds only when the terms are equal — which is usually what the question wants you to state.

FAQs

When can I use the infinite GP sum formula?

Only when the common ratio satisfies |r| < 1. Outside that range the series does not converge and the formula has no meaning, even though it will still produce a number if you substitute into it.

How do I insert n arithmetic means between two numbers?

Treat the two given numbers as the first and (n + 2)^th terms of an AP. Find d from that, then generate the means. The same logic with a common ratio inserts geometric means.

Why write three AP terms as a − d, a, a + d?

Because their sum is 3a, which removes d immediately. It converts a three-unknown problem into a one-unknown problem and is the fastest route through most "find the numbers" questions.

Is this chapter worth much in the board paper?

It carries a modest weight but has an unusually high return on time, because the formulas are few and the question types repeat closely from year to year.

Need help with this chapter?

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