A) \[\frac{16}{35}\]
B) \[\frac{11}{8}\]
C) \[\frac{35}{16}\]
D) \[\frac{9}{16}\]
Correct Answer: C
Solution :
Let \[S=1+\frac{4}{5}+\frac{7}{{{5}^{2}}}+\frac{10}{{{5}^{3}}}+....\] \[\frac{1}{5}S=\frac{1}{5}+\frac{4}{{{5}^{2}}}+\frac{7}{{{5}^{3}}}+......\] On subtracting, \[\left( S-\frac{1}{5}S \right)=1+\frac{3}{5}+\frac{3}{{{5}^{2}}}+\frac{3}{{{5}^{3}}}+....\] \[\Rightarrow \]\[\frac{4}{5}S=1+\frac{3}{5}\left[ 1+\frac{1}{5}+\frac{1}{{{5}^{2}}}+....... \right]\] \[=1+\frac{3}{5}\left[ \frac{1}{1-1/5} \right]=1+\frac{3}{5}\times \frac{5}{4}\] \[\Rightarrow \] \[\frac{4}{5}S=1+\frac{3}{4}=\frac{7}{4}\] \[\Rightarrow \] \[S=\frac{7}{4}\times \frac{5}{4}=\frac{35}{16}\]
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