Sequences & Series for JEE Maths
Sequences & Series builds on the AP/GP foundation from Class 10 and extends into powerful summation techniques used throughout JEE. This topic appears in almost every paper and is closely connected to Limits, Calculus, and Binomial Theorem. The problems range from straightforward formula application to tricky telescoping and method-of-differences questions.
Prerequisites: Class 10 Arithmetic Progressions
What JEE Tests
Key areas:
- AP: nth term, sum of n terms, arithmetic mean between two numbers
- GP: nth term, sum of n terms, sum to infinity (|r| < 1), geometric mean
- HP: relationship with AP (reciprocals form AP)
- AGP (Arithmetic-Geometric Progression): summation technique
- Special series: Σn, Σn², Σn³, and their formulas
- Method of differences for non-standard series
- Telescoping series
- AM-GM-HM inequality and its applications
Core Formulas
AP: a_n = a + (n-1)d, S_n = n/2[2a + (n-1)d] = n/2(first + last).
GP: a_n = ar^{n-1}, S_n = a(rⁿ - 1)/(r - 1). Sum to infinity: S∞ = a/(1-r) when |r| < 1.
Special sums: Σk = n(n+1)/2, Σk² = n(n+1)(2n+1)/6, Σk³ = [n(n+1)/2]².
AM-GM inequality: (a+b)/2 ≥ √(ab) for positive a,b. Equality when a = b. This is a powerful tool for finding maximum/minimum values without calculus.
Advanced Techniques
Method of differences: if the first differences of a sequence form an AP or GP, express the nth term using differences and sum.
Telescoping: rewrite each term as f(k) - f(k+1), so the sum collapses to f(1) - f(n+1). Partial fractions are often needed: 1/(k(k+1)) = 1/k - 1/(k+1).
AGP sum: for series like 1 + 2r + 3r² + 4r³ + ..., multiply by r and subtract to get a GP.
Key Formulas
Common Mistakes to Avoid
- Mistake: Using GP sum formula when r = 1 (it's undefined; use S_n = na instead)
- Mistake: Forgetting |r| < 1 condition for infinite GP sum
- Mistake: Confusing HP with 'harmonic series' — HP terms are reciprocals of AP terms
- Mistake: Not recognizing telescoping opportunities in summation problems
How to Practice This Topic
Week 1: AP and GP problems (15 each). Week 2: Special series summation and AM-GM inequality (15 problems). Week 3: Telescoping, method of differences, AGP (10 problems). These build on each other.
Formula Sheets
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