JEE Main Maths · Sequences & Series PYQ
JEE Main Sequences & Series Previous Year Questions (2002–2026)
Sequences & Series is a P1 Algebra chapter — ~5% JEE Main weightage with 1–2 questions per session. AP, GP, HP, AGP and AM-GM-HM inequality questions appear regularly. Mastery here accelerates Binomial Theorem and Probability downstream.
Sequences & Series PYQs from 2002 to 2026, tagged by sub-topic (AP, GP, telescopic, inequality) and difficulty. Every solution identifies the sequence type and applies the correct formula.
Sequences & Series at a Glance
Try 5 Real Sequences & Series PYQs
5 real JEE Main questions on Sequences & Series, straight from past papers. Try them, then check the answer.
- Q1JEE Main 2026 (Online) 21st January Evening Shift
The positive integer n, for which the solutions of the equation
$x(x+2) + (x+2)(x+4) + \cdots + (x+2n-2)(x+2n) $${}= \frac{8n}{3}$ are two consecutive even integers, is :
3
6
9
12
- Q2JEE Main 2025 (Online) 23rd January Morning Shift
If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
- $-120$
- $-1200$
- $-1080$
- $-1020$
- Q3JEE Main 2024 (Online) 30th January Morning Shift
Let $\alpha$${}=1^2+4^2+8^2+13^2+19^2+26^2+\ldots$ upto 10 terms and $\beta$${}=\sum\limits_{n=1}^{10} n^4$. If $4 \alpha-\beta$${}=55 k+40$, then k is equal to __________.
- Q4JEE Main 2023 (Online) 10th April Morning Shift
The sum of all those terms, of the arithmetic progression 3, 8, 13, ...., 373, which are not divisible by 3, is equal to ____________.
- Q5JEE Main 2022 (Online) 27th July Evening Shift
$ \frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}$ is equal to _____________.
Practise the full Sequences & Series set
Every past-paper question of this chapter, with solutions, timing and your accuracy tracked.
Key Sub-Topics & What's Tested
Arithmetic Progression (AP)
nth term a_n = a + (n-1)d, sum S_n = n/2 [2a + (n-1)d] = n/2 (a + l), properties, insertion of AMs.
Geometric Progression (GP)
nth term a_n = ar^(n-1), sum S_n = a(r^n - 1)/(r-1), infinite GP S_∞ = a/(1-r) for |r| < 1, insertion of GMs.
Harmonic Progression (HP)
Sequence whose reciprocals are in AP, harmonic mean, relationship with AP via reciprocal transformation.
AGP (Arithmetic-Geometric)
Terms of form (a + (n-1)d)·r^(n-1), sum formulas, applications in derivative of GP series.
AM ≥ GM ≥ HM Inequality
For positive reals, AM ≥ GM ≥ HM with equality iff all equal, weighted versions, applications in optimisation.
Sum of Special Series
Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]², sum of cubes, telescopic series techniques.
Telescopic Series
Series of form Σ[f(k+1) - f(k)] collapse to f(n+1) - f(1); partial fractions decomposition often unlocks telescopic form.
Insertion of Means
Inserting k AMs between a and b, k GMs between a and b, relationship A·H = G² for single means.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| AP nth Term / Sum | 20% | Given AP with sum of first 10 terms = 100, find 5th term. |
| GP nth Term / Sum | 20% | Sum of infinite GP = 3, first term = 2 — find common ratio. |
| AM-GM Inequality | 15% | Find minimum value of x + 1/x for x > 0 using AM-GM. |
| AGP Sum | 10% | Sum to infinity of 1 + 2x + 3x² + 4x³ + ... for |x| < 1. |
| Telescopic Series | 15% | Evaluate Σ 1/[k(k+1)] for k from 1 to n. |
| Mixed / Insertion | 20% | Four GMs between 2 and 54 — find the second GM. |
How to Solve Sequences & Series PYQs
- 1Identify the sequence type first. AP (constant difference), GP (constant ratio), HP (AP of reciprocals), AGP (AP × GP). Wrong identification breaks everything.
- 2For sum problems, use standard formulas. S_n = n/2(a + l) for AP; S_n = a(r^n - 1)/(r-1) for GP. Memorise, don't re-derive every time.
- 3AM-GM for optimisation: set up the inequality. Arithmetic Mean ≥ Geometric Mean with equality at AM = GM. Use for finding min/max of products and sums.
- 4Telescopic series: partial fractions. For Σ 1/[k(k+a)], decompose as (1/a)[1/k - 1/(k+a)] to unlock telescopic collapse.
- 5For infinite GP, check |r| < 1. S_∞ exists only if |r| < 1. If r ≥ 1, series diverges — don't apply S_∞ formula.
Common Mistakes That Cost Marks
- Using AM instead of HM for harmonic problems. Average speed over equal distances = HM of speeds, NOT AM. Classic trap.
- Forgetting (n-1) in nth term formula. a_n = a + (n-1)d, NOT a + nd. Off-by-one error is the #1 AP mistake.
- Applying AM-GM with negative terms. AM-GM works only for non-negative reals. If any term can be negative, split cases.
- Wrong infinite GP condition. Formula works only for |r| < 1. If r = 1 or r > 1, sum diverges.
- Miscounting AMs/GMs inserted. If k AMs are inserted between a and b, total sequence has k+2 terms (a, k AMs, b). Common off-by-one in insertion problems.
Related JEE Main Practice
Frequently asked questions
How many Sequences & Series PYQs should I solve?
Target 60–80 PYQs across 2010–2026. Given high template repeat (AP/GP sum, AM-GM) and 5% weightage, focused practice locks in reliable scores.
Is AGP heavily tested in JEE Main?
Moderately — 2-3 AGP PYQs per 10-year window. Usually as sum to infinity of an AGP with simple coefficients. Master the differentiation trick: AGP sum = differentiate GP.
Do I need to memorise sum formulas?
Yes — Σn, Σn², Σn³ formulas and AP/GP sum formulas are non-negotiable. Saves 90+ seconds per question vs deriving from scratch.
What's the most useful AM-GM application?
Finding minimum of a + b when ab = constant (or vice versa). PYQs wrap this in word problems — identify the constraint, apply AM ≥ GM.
How do I spot telescopic series?
Look for Σ of fractions like 1/[k(k+1)] or Σ of differences f(k+1) - f(k). If you can partial-fraction or recognize telescopic pattern, series collapses cleanly.
Does Sequences & Series connect with other chapters?
Yes — Binomial Theorem (sum of coefficients), Probability (expected value computations), Integration (Riemann sums). Strong fluency here helps downstream.
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P1 Algebra chapter. Chapter-wise PYQs on AP, GP, HP, AGP and AM-GM inequality with KaTeX solutions. Sign in in 30 seconds.
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