1. Tardigrade
  2. Question
  3. Mathematics
  4. The sum of the infinite terms of the series 1+(4/5)+(7/52)+(10/53)+... is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The sum of the infinite terms of the series $ 1+\frac{4}{5}+\frac{7}{{{5}^{2}}}+\frac{10}{{{5}^{3}}}+... $ is

Rajasthan PETRajasthan PET 2005

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} $