Question

The sum of an infinite geometric series is 12, and the first term is 8. Find the common ratio.

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{2}$
• D. $\frac{3}{4}$

Hint:

Use $\frac{a}{1 - r}$ for infinite GP sum and solve for $r$.

Answer 1

The Correct Option is A

We know that the sum of an infinite geometric series is given by the formula:
\[ S_{\infty} = \frac{a}{1 - r} \] where:
\(S_{\infty} =\) Sum to infinity = \(12\)
\(a =\) First term = \(8\)
\(r =\) Common ratio (to be found)
Substituting the given values:
\[ 12 = \frac{8}{1 - r} \] Multiply both sides by \((1 - r)\):
\[ 12(1 - r) = 8 \] Expand the left-hand side:
\[ 12 - 12r = 8 \] Rearranging:
\[ -12r = 8 - 12 \] \[ -12r = -4 \] Dividing both sides by \(-12\):
\[ r = \frac{-4}{-12} \] \[ r = \frac{1}{3} \] \[ \boxed{r = \frac{1}{3}} \]