Sum of GP
The sum of the first n terms of a GP (with r ≠ 1) is S_n = a(r^n − 1)/(r − 1). When r > 1 this form keeps the numbers positive; when r < 1 the equivalent S_n = a(1 − r^n)/(1 − r) is tidier. If r = 1 every term equals a, so the sum is simply n·a — a special case CAT sometimes slips in to catch the unwary. The fastest approach in the exam is to compute r^n first using powers you already know (2^10 = 1024, 3^4 = 81), then plug in. A useful manipulation: S_n(r − 1) = a(r^n − 1), which lets you solve for an unknown a, r or n without dividing early and accumulating fractions. Recognise the sum of a GP inside compound-interest annuities and inside questions that ask for 1 + 2 + 4 + ... + 2^k or 1 + 3 + 9 + ... patterns.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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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 |