Find the sum to infinity of the geometric series whose first term is $12$ and common ratio is $\frac{1}{4}$.

Step-by-step solution

1. Let $a=12$ be the first term and $r=\frac{1}{4}$ be the common ratio.
2. Check convergence: $|r|<1$, so the series converges to $S_{\infty }=\frac{a}{1-r}$.
3. Substitute $a=12$ and $r=\frac{1}{4}$: $S_{\infty }=\frac{12}{1-\frac{1}{4}}=\frac{12}{\frac{3}{4}}$.
4. Divide by $\frac{3}{4}$ which is same as multiplying by $\frac{4}{3}$: $12\times \frac{4}{3}=16$.
5. Answer: $16$.

Concept tested: Geometric series to infinity

Published 16 October 2026 · How this solution was prepared and checked