JEE Main 2016 (23)

Say $\alpha$ is the first term of the given infinite geometric progression. The given information can be mathematically framed as :- $T_n\,=\,4(S_∞\,-\,S_n)$ , where $T_n, S_n$ and $S_∞$ denote, respectively, the $n^{th}$ term, sum of first $n$ terms and sum of all the terms of the given infinite geometric progression. Now, $T_n\,=\,4(S_∞\,-\,S_n)\ \implies \alpha \cdot r^{n-1}\,=\,4 \left(\frac{\alpha}{1-r}\,-\,\frac{\alpha \cdot [1-r^{n}]}{1-r}\right)$ .

Simplifying this gives $r\,=\,\frac{1}{5}$ . So, $10 \cdot r \,=\, 2$