Sequences & Series
A sequence is an ordered list of numbers; a series is what you get when you add them. CAT almost never asks you to plod through "add these forty terms" — it hands you a sum that looks frightening and quietly rewards the student who recognises the pattern and reaches for the right identity. Three engines do most of the heavy lifting: the special sums (1+2+...+n, the sum of squares, the sum of cubes), the telescoping trick where a clever split makes the middle of a sum collapse, and the arithmetic-geometric progression (AGP) where each term is an AP term times a GP term. Layered on top are simple recurrences — the Fibonacci family and "each term depends on the previous one or two" definitions — which CAT loves because they punish guessing and reward careful term-by-term reasoning or a closed form. This chapter builds all three. You will learn to spot which engine a problem needs, to compute the standard sums in one line, to manufacture telescoping by partial fractions, to crack an AGP with the S − rS shift, and to tame a recurrence by either unrolling it or finding its pattern. The payoff is speed: a problem that looks like a three-minute slog becomes a thirty-second identity.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Σk = n(n+1)/2, Σk² = n(n+1)(2n+1)/6, Σk³ = [n(n+1)/2]². Memorise all three; the cube sum is just the square of the natural-number sum.
- For any "sum of a polynomial in k", split by linearity (Σ(ak² + bk + c) = aΣk² + bΣk + cn) and apply the standard identities — never add term by term.
- See a product in a denominator? Suspect telescoping. 1/[k(k+1)] = 1/k − 1/(k+1); the middle cancels and only the ends survive.
- General telescoping: 1/[k(k+d)] = (1/d)[1/k − 1/(k+d)]. The factor 1/d in front is the part students forget.
- For an AGP (AP term × GP term, e.g. n·rⁿ), use the S − rS shift: subtract r·S from S to collapse the stagger into a GP.
- For any recurrence, unroll three or four terms first — many CAT recurrences are periodic or have an obvious closed pattern, so you rarely need heavy theory.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Confusing the sum of squares n(n+1)(2n+1)/6 with the square of the sum [n(n+1)/2]² — they are different (the second equals the sum of cubes).
- In telescoping 1/[k(k+d)], forgetting the leading factor 1/d, which scales the entire answer.
- Treating the partial sum up to n as the nth term (or vice versa) — always ask whether the pattern describes a term or a running total.
- Adding successive Fibonacci or recurrence terms by hand and slipping by one index — losing track of whether the count starts at F1 or F0.
- On an AGP, summing it as if it were a plain GP and ignoring the arithmetic multiplier — you must use the S − rS shift, not a/(1−r) alone.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Sequences & Series when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Special sums (first n terms)
| Sum of first n naturals | 1+2+...+n = n(n+1)/2 |
|---|---|
| Sum of first n squares | 1²+2²+...+n² = n(n+1)(2n+1)/6 |
| Sum of first n cubes | 1³+2³+...+n³ = [n(n+1)/2]² |
| Sum of first n odd numbers | 1+3+5+...+(2n−1) = n² |
| Sum of first n even numbers | 2+4+...+2n = n(n+1) |
| Key link: cubes = (sum)² | Σk³ = (Σk)² = [n(n+1)/2]² |
Telescoping, AGP & recurrences
| Telescoping split | 1/[k(k+1)] = 1/k − 1/(k+1) |
|---|---|
| General telescoping | 1/[k(k+d)] = (1/d)[1/k − 1/(k+d)] |
| AGP sum to infinity (|r|<1) | S∞ = a/(1−r) + dr/(1−r)² |
| AGP finite sum method | Compute S − rS to collapse to a GP |
| Linear recurrence (Fibonacci) | F(n) = F(n−1) + F(n−2) |
| Sum of first n Fibonacci | F1+F2+...+Fn = F(n+2) − 1 |