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Sequences and series is often taught as a list of formulas: sum of an AP, sum of a GP, sum of squares, sum of cubes. Students memorise them, then freeze when a question changes the pattern slightly. Each of these formulas comes from a trick that takes three lines. If you know the trick, you can rebuild the formula and also handle the variations that JEE likes to set.
What does the syllabus cover?
The JEE (Main) 2026 syllabus lists "Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers, Relation between A.M and G.M." The JEE (Advanced) 2026 brochure adds sums of finite arithmetic and geometric progressions, infinite geometric series, and "sum of the first n natural numbers, sums of squares and cubes of the first n natural numbers". NCERT's Class 11 chapter on sequences and series covers APs, GPs and the AM-GM relation. Check the documents for your exam year.
Why derive instead of memorise?
In two experiments with school students learning decimals, Rittle-Johnson, Siegler and Alibali (2001) found that conceptual and procedural knowledge "develop iteratively": understanding predicted gains in skill, and gains in skill predicted better understanding. Better problem representation was one link between them. Deriving a formula is a small version of that cycle. You use the procedure and see why it works, and the next variation of the problem looks familiar instead of new.
The AP sum: pair the terms
Write the sum forwards and backwards:
S = a + (a + d) + … + l S = l + (l − d) + … + a
Each column adds to a + l, and there are n columns, so 2S = n(a + l) and S = n(a + l)/2. With l = a + (n − 1)d, this becomes S = (n/2)[2a + (n − 1)d].
Example. The two-digit multiples of 7 run from 14 to 98. The count is (98 − 14)/7 + 1 = 13, so the sum is 13 × (14 + 98)/2 = 13 × 56 = 728.
The GP sum: subtract rS
S = a + ar + … + arⁿ⁻¹ rS = ar + … + arⁿ⁻¹ + arⁿ
Subtracting, S − rS = a − arⁿ, so S = a(1 − rⁿ)/(1 − r) for r ≠ 1. When |r| < 1, rⁿ shrinks to zero and the infinite sum is a/(1 − r).
Example. To insert two geometric means between 2 and 54, write 54 = 2r³, so r³ = 27 and r = 3. The means are 6 and 18.
The same subtraction handles a series like 1/2 + 2/4 + 3/8 + …, where each term is k/2ᵏ. Subtracting half the series from itself leaves a plain GP, 1/2 + 1/4 + 1/8 + … = 1, so half the sum is 1 and the sum is 2. If a series has an arithmetic part times a geometric part, try multiplying by r and subtracting.
The sum of squares: telescope
Write (k + 1)³ − k³ = 3k² + 3k + 1 for k = 1 to n and add. The left side telescopes to (n + 1)³ − 1. The right side is 3Σk² + 3n(n + 1)/2 + n. Solving gives Σk² = n(n + 1)(2n + 1)/6. The same method with fourth powers gives Σk³ = [n(n + 1)/2]².
Worked example. Find 1·2 + 2·3 + … + n(n + 1), and its value for n = 10.
Each term is k² + k, so the sum is n(n + 1)(2n + 1)/6 + n(n + 1)/2 = n(n + 1)(2n + 4)/6 = n(n + 1)(n + 2)/3. For n = 10, this is 10 × 11 × 12/3 = 440. Check: Σk² to 10 is 385 and Σk is 55, and 385 + 55 = 440.
How does AM ≥ GM give minimum values?
For positive a and b, (√a − √b)² ≥ 0 expands to (a + b)/2 ≥ √(ab), with equality only when a = b.
Example. For x > 0, find the least value of x + 9/x. The AM of x and 9/x is at least √9 = 3, so x + 9/x ≥ 6, with equality when x = 9/x, that is, x = 3.
Which mistakes cost the most marks?
Miscounting terms. From 14 to 98 in steps of 7 there are 13 terms, not 12. Use (last − first)/d + 1.
Using a/(1 − r) when |r| ≥ 1. The infinite sum exists only for |r| < 1.
Using AM-GM with negative numbers, or forgetting to check that equality can actually happen.
Writing Σk² = (Σk)². For n = 10, that is 385 against 3025.
Losing the last term when telescoping.
A practice routine
Derive from blank paper (once a week): AP sum, GP sum, Σk² and the infinite GP, without notes. Time yourself; aim for two minutes each.
General-term drill (three times a week): five series where you must write the k-th term and split it before summing.
AM-GM set (weekly): five minimum-value problems. For each, write the equality condition.
Explain one step aloud. The self-explanation technique works well for derivations. Keep the derivations, not just the results, on your own formula sheet.
If a series will not split and you cannot see why, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can compare its first step with yours.
Tonight, derive Σk² from the telescoping identity without looking, then use it to find 1² + 2² + … + 20².
