alleryThe sum of the first $n$ terms of a GP with first term $a$ and common ratio $r \neq 1$ is
A$S_n = a\,(r^n - 1)/(r - 1)$
B$S_n = a\,(1 + r)^n$
C$S_n = an\,r$
D$S_n = a/(1 - r)$
Answer & Solution
Correct answer: A. $S_n = a\,(r^n - 1)/(r - 1)$
1. syllabus derives the GP sum.
2. Write: $S_n = a + ar + ar^2 + \ldots + ar^{n-1}$.
3. Multiply both sides by $r$: $r S_n = ar + ar^2 + \ldots + ar^n$.
4. Subtract: $S_n - r S_n = a - a r^n$, so $S_n (1 - r) = a(1 - r^n)$.
5. Solve: $S_n = \dfrac{a(1 - r^n)}{1 - r} = \dfrac{a(r^n - 1)}{r - 1}$. Same formula written two ways.
6. Option B is wrong (binomial-style). Option C is the AP product. Option D is the INFINITE GP sum (only when $|r| < 1$).
_Source: syllabus Mathematics, Ch 8, §8.3 (GP sum), p. 6–7._
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