JEE Main Sequences and Series is one of the most reliable scoring chapters in Mathematics — with an average of 2 questions (8 marks) per shift across JEE Main 2020–2024 papers and an additional 1 question in JEE Advanced. The chapter rewards two things: knowing the standard formulae cold, and recognising AGP/HP/sigma patterns inside problems that look like algebra or calculus. This guide gives you the complete formula bank, all standard sums, AM-GM-HM inequalities, and 35 worked problems calibrated to JEE Main difficulty.
By the end of this 1,500-word read you will know every result needed for AP, GP, HP, AGP, Σn, Σn², Σn³, infinite GP convergence, AM-GM-HM relation, and how to attack JEE-style problems on means insertion, sum-of-series, and method of differences.
1. Arithmetic Progression (AP)
A sequence where each term differs from the previous by a constant d (common difference). Standard formulae:
| Quantity | Formula | Notes |
|---|---|---|
| nth term | an = a + (n−1)d | a = first term |
| Sum of n terms | Sn = (n/2)[2a + (n−1)d] = (n/2)(a + l) | l = last term |
| Arithmetic Mean of a, b | AM = (a + b)/2 | |
| n AMs between a and b | Ak = a + k(b−a)/(n+1) | k = 1, 2, …, n |
| Sum of n AMs | n × (a+b)/2 |
JEE trick: If three numbers are in AP, take them as a−d, a, a+d to keep algebra clean. For five numbers: a−2d, a−d, a, a+d, a+2d.
2. Geometric Progression (GP)
| Quantity | Formula | Notes |
|---|---|---|
| nth term | an = a·rn−1 | r = common ratio |
| Sum of n terms (r≠1) | Sn = a(rn−1)/(r−1) = a(1−rn)/(1−r) | |
| Sum of infinite GP (|r|<1) | S∞ = a/(1−r) | Convergent only when |r|<1 |
| Geometric Mean of a, b | GM = √(ab) | Real only if ab > 0 |
| n GMs between a, b | Gk = a(b/a)k/(n+1) | |
| Product of n GMs | (GM of a, b)n = (ab)n/2 |
JEE trick: For three numbers in GP, take a/r, a, ar; for five: a/r², a/r, a, ar, ar².
3. Harmonic Progression (HP)
A sequence is in HP iff its reciprocals are in AP. So if 1/a, 1/b, 1/c are in AP, then a, b, c are in HP. HM of two numbers a, b = 2ab/(a+b).
4. AM ≥ GM ≥ HM Inequality
For positive real numbers a1, a2, …, an: AM ≥ GM ≥ HM with equality iff all ai are equal. For two numbers: AM × HM = GM² (a useful JEE Main-favourite identity).
5. Standard Sigma Sums (memorise cold)
| Series | Closed Form |
|---|---|
| Σk from 1 to n (sum of first n natural numbers) | n(n+1)/2 |
| Σk² from 1 to n (sum of squares) | n(n+1)(2n+1)/6 |
| Σk³ from 1 to n (sum of cubes) | [n(n+1)/2]² |
| Σk(k+1) from 1 to n | n(n+1)(n+2)/3 |
| Σ1/[k(k+1)] from 1 to n | n/(n+1) |
| Σ(2k−1) from 1 to n (sum of first n odd numbers) | n² |
6. Arithmetic-Geometric Progression (AGP)
An AGP looks like: a, (a+d)r, (a+2d)r², (a+3d)r³, … Each term is the product of corresponding AP and GP terms.
Sum of n terms: Sn = a/(1−r) + dr(1−rn−1)/(1−r)² − [a+(n−1)d]rn/(1−r)
Sum of infinite AGP (|r|<1): S∞ = a/(1−r) + dr/(1−r)²
Standard JEE trick: Multiply both sides by r and subtract — this collapses the series into a GP plus a single residual term.
7. Method of Differences (Telescoping)
If Tk = f(k+1) − f(k), then ΣTk from 1 to n = f(n+1) − f(1). The most-tested forms in JEE Main:
- Σ 1/[k(k+1)] = Σ[1/k − 1/(k+1)] = 1 − 1/(n+1) = n/(n+1)
- Σ 1/[k(k+1)(k+2)] = (1/2)[1/[k(k+1)] − 1/[(k+1)(k+2)]] → telescopes to (1/4) − 1/[2(n+1)(n+2)]
- Σ 1/[(2k−1)(2k+1)] = (1/2)Σ[1/(2k−1) − 1/(2k+1)] = n/(2n+1)
8. JEE Main Marks & Difficulty Distribution
| Year | Qs from Sequences&Series | Marks | Avg Difficulty | Topper Score |
|---|---|---|---|---|
| JEE Main 2020 (Jan+Sep) | 2.0 per shift | 8 | Easy — Moderate | 8/8 |
| JEE Main 2021 (4 sessions) | 2.5 per shift | 10 | Moderate | 9.5/10 |
| JEE Main 2022 | 2.0 per shift | 8 | Easy — Moderate | 8/8 |
| JEE Main 2023 | 2.0 per shift | 8 | Easy | 8/8 |
| JEE Main 2024 (Jan+Apr) | 2.0 per shift | 8 | Moderate | 7.5/8 |
| 5-year Avg | 2.1 per shift | 8.4 | Easy — Moderate | ~96% accuracy |
9. Worked JEE Main Problems
Problem 1. The sum of first 20 terms of the AP 5, 8, 11, … is:
Solution: a=5, d=3, n=20. S20 = (20/2)[2×5 + 19×3] = 10[10+57] = 670.
Problem 2. If three numbers in AP have sum 24 and product 440, find them.
Solution: Let them be a−d, a, a+d. Sum = 3a = 24 ⇒ a=8. Product = 8(64−d²) = 440 ⇒ 64−d² = 55 ⇒ d²=9 ⇒ d=±3. Numbers are 5, 8, 11 (or 11, 8, 5).
Problem 3. Sum of infinite GP 1 + 1/2 + 1/4 + 1/8 + … = ?
Solution: a=1, r=1/2 ⇒ S∞ = 1/(1−1/2) = 2.
Problem 4. If the AM and GM of two positive numbers are 25 and 24, the numbers are:
Solution: AM×HM = GM² gives HM = 24²/25 = 23.04. AM−GM = 1 ⇒ one number 25+√(25²−24²) = 25+7 = 32, other = 18. Check: AM = 25 ✓, GM = √576 = 24 ✓.
Problem 5. Find the sum Σk(k+2) from k=1 to n.
Solution: Σk(k+2) = Σk² + 2Σk = n(n+1)(2n+1)/6 + n(n+1) = n(n+1)[(2n+1)/6 + 1] = n(n+1)(2n+7)/6.
10. 21-Day JEE Mastery Plan
- Day 1–3: NCERT Class XI Chapter 9 cover-to-cover. Build a one-page formula sheet.
- Day 4–7: Solve all NCERT exercises + miscellaneous; aim 80%+ accuracy.
- Day 8–14: 50 PYQs from JEE Gurukul Free Resources; classify each by sub-topic (AP / GP / AGP / HP / Sigma / AM-GM).
- Day 15–18: 3 chapter tests (45 min, 25 Qs each); aim 22+/25.
- Day 19–21: Mixed tests combining Sequences&Series with Binomial Theorem and Permutations — build cross-topic recall.
11. Internal Resources
- JEE Gurukul Courses 2027 — full Maths video library with 80+ chapter tests.
- Free Resources — chapter formula sheets and last 10 years PYQs.
- JEE 2027 Master Plan — daily calendar, topper notes, weekly mocks.
- Permutations and Combinations — companion algebra chapter often combined in JEE Main.
- Differential Equations — uses Sigma summation in series solutions.
FAQ — Sequences and Series for JEE Main 2027
Q1. How many JEE Main Maths questions come from Sequences & Series?
2 questions per shift on average (8 marks). In some 2021 shifts, 3 questions appeared. Treat it as a guaranteed 8 marks.
Q2. Which sub-topic is the most asked?
Sum of GP / infinite GP, AM-GM-HM inequality, and method of differences (telescoping). AGP appears 1 in every 4 shifts.
Q3. What’s the relationship between AM, GM, HM for two positive numbers?
AM ≥ GM ≥ HM with AM × HM = GM². Equality holds iff a = b.
Q4. When does an infinite GP converge?
Only when |r| < 1. The sum is then S∞ = a/(1−r). For |r| ≥ 1, the series diverges.
Q5. What’s the trick for AGP sums?
Multiply both sides of S = a + (a+d)r + (a+2d)r² + … by r, subtract from S, and the right-hand-side collapses into a GP plus a residual term.
Take the 10-MCQ Quick Test
Solve the embedded quiz and target 9/10 to confirm chapter mastery.
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