Geometric Progression
A geometric progression (GP) is a sequence in which every term after the first is obtained by multiplying the previous one by a fixed number called the common ratio r. So 3, 6, 12, 24, ... is a GP with r = 2, and 81, 27, 9, 3, ... is a GP with r = 1/3. Where an arithmetic progression grows by adding a constant, a GP grows by repeated multiplication, which is exactly the maths behind compound interest, population growth and depreciation — topics CAT loves to disguise as sequence problems. This chapter builds the three tools you actually need in the exam: the nth-term formula a·r^(n−1) to jump to any term without listing the sequence, the finite-sum formula a(r^n−1)/(r−1) to add a block of terms in one step, and the infinite-sum formula a/(1−r) for the surprisingly common case |r| < 1, where an endless sum settles on a finite number. Along the way we use the geometric mean √(ab) and the symmetric-terms trick (a/r, a, ar) that turns ugly product-and-sum conditions into clean equations. Every section gives the fast CAT method, worked examples of rising difficulty, and the traps that cost marks.
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.
- To find r from two terms, divide them: a_m/a_n = r^(m−n). This wipes out the unknown first term a.
- For three numbers in GP, take them as a/r, a, ar — their product is a³, so the middle term is the cube root of the product.
- Memorise small powers (2^10 = 1024, 3^4 = 81, 3^6 = 729) so the sum formula needs no long multiplication.
- Recurring decimal 0.x̄ = x/9, 0.xȳ = xy/99 — both come straight from the infinite-GP sum a/(1 − r) with r a power of 1/10.
- Infinite sum exists only when |r| < 1. If |r| ≥ 1, the answer is "does not converge" — a common CAT trap option.
- Sum of squares of a GP is a GP with ratio r²; sum to infinity = a²/(1 − r²). Same trick for products of consecutive terms.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Using a·r^n instead of a·r^(n−1) for the nth term — off by one factor of r.
- Applying the infinite-sum formula when |r| ≥ 1, where the series actually diverges.
- Forgetting the r = 1 case: then the sum is n·a, not a(r^n − 1)/(r − 1) (division by zero).
- Mishandling a negative common ratio — terms alternate in sign and (−r)^n flips with parity of n.
- Confusing the geometric mean √(ab) with the arithmetic mean (a + b)/2 when a question says "mean".
📈 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 Geometric Progression 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
Core GP formulas
| nth term | a_n = a·r^(n−1) |
|---|---|
| Sum of n terms (r ≠ 1) | S_n = a(r^n − 1)/(r − 1) |
| Sum of n terms (|r| < 1 form) | S_n = a(1 − r^n)/(1 − r) |
| Infinite sum (|r| < 1) | S_∞ = a/(1 − r) |
| Geometric mean of a and b | GM = √(ab) |
CAT power-tools
| Three terms in GP | a/r, a, ar (product = a³) |
|---|---|
| Ratio of two terms | a_m / a_n = r^(m − n) |
| n GMs between a and b | common ratio r = (b/a)^(1/(n+1)) |
| Sum × (r − 1) | S_n(r − 1) = a(r^n − 1) |
| Each term squared | a², a²r², a²r⁴, ... is a GP with ratio r² |