Question

The sum of an infinite GP is x and the common ratio r is such that $|r|<1$. If the first term of the GP is 2, then which one of the following is correct ?

• A. $-1<x<1$
• B. $\infty <x<1$
• C. $1<x<\infty$
• D. None of the above

Answer 1

The correct option is C $1<x<\infty$

Given: $\left|r\right|<1$

Case 1: $r>0$

$\Rightarrow 0<1-r<1$

$\Rightarrow 1<\frac{1}{1-r}<\infty$

$x=\frac{2}{1-r}$

$2<x<\infty$

Case 2: $r<0$

$\Rightarrow 1<1-r<2$

$\Rightarrow 0.5<\frac{1}{1-r}<1$

$\Rightarrow 1<\frac{2}{1-r}<2$

Hence, on taking the union of the 2 cases, we get

$1<x<\infty$