Question

Find the least number of years for which an annuity Rs 1000 must run in order that its amount just exceeds Rs 16000 at 5% pa. compounded annually.

• A. $12$
• B. $9$
• C. $2$
• D. $25$

Answer 1

The correct option is A $12$

$P=4000,r=5\%$

$A={\sum }_{n=1}^{n_0}P{(1+\frac{r}{100})}^n={\sum }_{n=1}^{n_0}1000{(1+\frac{5}{100})}^n>1600={\sum }_{n=1}^{n_0}{(1+\frac{5}{100})}^n>16$

Since it is a sum of a infinite GP. We can write it as a

$\frac{({1.05}^{n_0}-1)}{(1.05-1)}\times 1.05>16$

$⟹(1.05{)}^{n_0}>1.7619$

Taking log on both sides

$⟹\log (1.05{)}^{n_0}>\log 1.7619$

$⟹n_0>\frac{\log \left(1.7619\right)}{\log \left(1.05\right)}$

$⟹n_0>11.6$

$⟹n_0=12$

Hence, least no. of years$=12$.