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Q.
The sum (upto two decimal places) of the infinite series $\frac{7}{17}+\frac{77}{1 7^{2}}+\frac{777}{1 7^{3}}+.......$ is
NTA AbhyasNTA Abhyas 2020Sequences and Series
Solution:
$S=\frac{7}{17}+\frac{77}{1 7^{2}}+\frac{777}{1 7^{3}}+.......$
$\frac{S}{17}=\frac{7}{17^{2}}+\frac{77}{17^{3}}+\frac{777}{17^{4}}+........$
Subtracting both the above results, we get,
$\frac{16}{17}S=\frac{7}{17}+\frac{70}{17^{2}}+\frac{700}{17^{3}}+.......$
$\left(I n f i n i t e \, G . P . w i t h \, r \, = \frac{10}{17}\right)$
$\frac{16}{17}S=\frac{\frac{7}{17}}{1 - \frac{10}{17}}=\frac{\frac{7}{17}}{\frac{7}{17}}=1$
$S=\frac{17}{16}$