Question

Evaluate: $\underset{n\to \infty }{lim}\left[\frac{7}{10}+\frac{29}{{10}^2}+\frac{133}{{10}^3}+\dots \dots +\frac{5^n+2^n}{{10}^n}\right]$

• A. $\frac{3}{4}$
• B. $2$
• C. $\frac{5}{4}$
• D. $\frac{1}{2}$

Answer 1

The correct option is C $\frac{5}{4}$

$\underset{n\to \infty }{lim}\sum _{k=1}^n\frac{5^k+2^k}{{10}^k}$

$=\underset{n\to \infty }{lim}\sum _{k=1}^n(\frac{5}{10}{)}^k+(\frac{2}{10}{)}^k$

$=\sum _{k=1}^{\infty }{\left(\frac{1}{2}\right)}^k+\sum _{k=1}^{\infty }{\left(\frac{1}{5}\right)}^k$

$=\frac{\frac{1}{2}}{1-\frac{1}{2}}+\frac{\frac{1}{5}}{1-\frac{1}{5}}$ (use G.P formula for summation of infinite series)

$=1+\frac{1}{4}=\frac{5}{4}$