Question

The sum of an infinite geometric series is 162 and the sum of its first n terms is 160. The inverse of its common ratio is an integer, then how many values of the common ratio (r) are possible?

• A. 1
• B. 2
• C. 4
• D. 5

Answer 1

The correct option is D 5

$\frac{a}{1-r}=162,\frac{a(1-r^n)}{1-r}=160\therefore 1-r^n=\frac{80}{81}\Rightarrow r^n=\frac{1}{81}Now,since\frac{1}{r}ϵX\therefore \frac{1}{r^n}=81\Rightarrow \frac{1}{r}=3^{4/n}\Rightarrow \therefore n=1,2,4$
Here 1, 2, 4 are the factors of 4.
$\begin{array}{ccc}n& r& a\\ 1& \frac{1}{81}& 160\\ 2& \frac{1}{9},-\frac{1}{9}& 144and180\\ 4& \frac{1}{3,}-\frac{1}{3}& 108and216\end{array}$
Hence choice (d) is correct.