Mathematics - Sequence & Series Question with Solution | TestHub

Let $S=1+\frac{5}{6}+\frac{12}{6^2}+\frac{22}{6^3}+\frac{35}{6^4}+\dots$ .....(i) $\frac{S}{6}=\frac{1}{6}+\frac{5}{6^2}+\frac{12}{6^3}+\frac{22}{6^4}+\dots$ ....(ii) On subtracting (i) from (ii) $\frac{5}{6}S=1+\frac{4}{6}+\frac{7}{6^2}+\frac{10}{6^3}+\frac{13}{6^4}+\cdots$ $\frac{5}{36}S=\frac{1}{6}+\frac{4}{6^2}+\frac{7}{6^3}+\frac{10}{6^4}+\frac{13}{6^5}+\cdots$ on subtraction $\frac{25}{36}S=1+\frac{3}{6}+\frac{3}{6^2}+\frac{3}{6^3}+\dots =\frac{8}{5}$ $S=\frac{8}{5}\times \frac{36}{25}=\frac{288}{125}$