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Q.
The value of $\displaystyle\sum_{n=1}^{\infty} \frac{5^n-2^n}{10^n}$ is equal to
Sequences and Series
Solution:
$\displaystyle\sum_{n=1}^{\infty} \frac{5^n-2^n}{10^n}=\displaystyle\sum_{n=1}^{\infty}\left(\frac{5^n}{10^n}-\frac{2^n}{10^n}\right)=\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n-\left(\frac{1}{5}\right)^n$
the difference between two infinite geometric sequences
$=\frac{\frac{1}{2}}{1-\frac{1}{2}}-\frac{\frac{1}{5}}{1-\frac{1}{5}}=1-\frac{1}{4}=\frac{3}{4}$