Sequences and Series
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Direct answer
For an arithmetic progression, the nth term is a + (n - 1)d and the sum of n terms is (n/2)[2a + (n - 1)d] = (n/2)(first + last). For a geometric progression, the nth term is a r^(n-1), the sum of n terms is a(r^n - 1)/(r - 1) for r not equal to 1, and the infinite sum is a/(1 - r) when |r| < 1. Almost every JEE question reduces to identifying the pattern, writing the correct Sn and using the three standard power sums.
What you must remember
- AP: an = a + (n - 1)d; Sn = (n/2)[2a + (n - 1)d] = (n/2)(a + l); arithmetic mean of a and b is (a + b)/2.
- GP: an = a r^(n - 1); Sn = a(r^n - 1)/(r - 1), r not 1; S infinity = a/(1 - r) for |r| < 1; geometric mean of two positive numbers is sqrt(ab); AM >= GM, with equality when the numbers are equal.
- HP: reciprocals form an AP; harmonic mean HM = 2ab/(a + b); for two numbers, GM^2 = AM × HM, so AM >= GM >= HM.
- Standard power sums: sum of first n natural numbers = n(n + 1)/2; sum of squares = n(n + 1)(2n + 1)/6; sum of cubes = [n(n + 1)/2]^2.
- Telescoping: 1/(n(n + 1)) = 1/n - 1/(n + 1), so the sum of the first n terms collapses to n/(n + 1).
- Arithmetico-geometric series like a + (a + d) r + (a + 2d) r^2 + ... : multiply the sum by r, subtract from the original, and solve — the shift-and-subtract method.
- Method of differences: when successive terms of a series have differences that are constant or in an AP, Tn is a polynomial in n; assume Sn = An^3 + Bn^2 + Cn and fit with S1, S2, S3.
Common confusion
The trap that never retires is mixing up Sn with the nth term. When a question gives the sum of the first n terms, the nth term is an = Sn - S(n - 1) — not the expression you read off Sn. Students also quote the infinite-GP sum without checking |r| < 1, and apply AM-GM to numbers that are not all positive. Each of these three slips converts an easy question into a confidently wrong answer.
Exam-focused takeaway
JEE Main leans on direct AP-GP sums, means and the power sums, usually as numerical-value questions that reward clean algebra. JEE Advanced prefers the machinery: telescoping with partial fractions, arithmetico-geometric sums, sequences defined by recurrences, and inequalities proved through AM-GM. Classification first, formula second.
Frequently asked questions
When does an infinite GP converge?
Only when |r| < 1; the sum is then a/(1 - r). For |r| >= 1 the partial sums grow without bound or oscillate.
How do I find the nth term from a given sum formula?
Use an = Sn - S(n - 1) for n >= 2, and check a1 = S1 separately.
What relates AM, GM and HM for two numbers?
GM^2 = AM × HM, and AM >= GM >= HM with equality only when the two numbers are equal.
Which formula gives the sum of the first n squares?
n(n + 1)(2n + 1)/6 — it appears constantly alongside n(n + 1)/2 and [n(n + 1)/2]^2.
Is 1 + 1/2 + 1/3 + ... a geometric series?
No. It is the harmonic series with reciprocals in AP; its partial sums grow without bound and no GP formula applies.