• # question_answer ${{n}^{th}}$ term of the series$1+\frac{4}{5}+\frac{7}{{{5}^{2}}}+\frac{10}{{{5}^{3}}}+........$will be A) $\frac{3n+1}{{{5}^{n-1}}}$ B) $\frac{3n-1}{{{5}^{n}}}$ C) $\frac{3n-2}{{{5}^{n-1}}}$ D) $\frac{3n+2}{{{5}^{n-1}}}$

This series is clearly an A.G., the corresponding A.P. is $1+4+7+10+.........$having ${{n}^{th}}$term $=3n-2$ and corresponding G.P. is  $1+\frac{1}{5}+\frac{1}{{{5}^{2}}}+.........$ having ${{n}^{th}}$term $=\frac{1}{{{5}^{n-1}}}$ Hence required ${{n}^{th}}$term of the series is $\frac{3n-2}{{{5}^{n-1}}}$. Trick: Check by putting $n=1,\ 2$ in alternates.