Find the sum to infinity of the geometric series whose first term is $5$ and common ratio is $\frac{2}{3}$ .

medium MCQ 2 marks Class 11 · Mathematics · Sequences and Series

Answer

Option C: 15/1

The correct answer is Option C: 15/1 — Thus the sum to infinity is $15$ .

Step-by-step solution

Check convergence: $∣ r ∣ = \frac{2}{3} < 1$ , so the series converges.
Use $S_{\infty} = \frac{a}{1 - r}$ with $a = 5$ and $r = \frac{2}{3}$ .
Compute $S_{\infty} = \frac{5}{1 - \frac{2}{3}} = \frac{5}{\frac{1}{3}} = 5 \cdot 3 = 15$ .
Thus the sum to infinity is $15$ .

Common mistake: Using

S_{n}

formula for infinite sum when

∣ r ∣ \geq 1

or miscomputing

1 - r

Concept tested: Geometric series convergence