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.
The formulas, and the condition each one needs
| Quantity | Formula | Condition or trap |
|---|---|---|
| nth term of an AP | an = a + (n − 1)d | It is (n − 1), not n |
| Sum of n terms of an AP | Sn = n/2 [2a + (n − 1)d] | Or n/2 (a + l) when the last term is known |
| nth term of a GP | an = arn−1 | Again (n − 1) |
| Sum of n terms of a GP | Sn = a(rn − 1)/(r − 1) | Undefined at r = 1; use Sn = 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.
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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