Rohit invests Rs 15625 for three years at a certain rate of interest, compounded annually. At the end of one year, it amounts to Rs 16875. Calculate: (i) the rate of interest per annum. (Ii) the interest accrued in the second year. (iii) the amount due to him at the end of third year.

(iv) the interest earned in the third year. (v) the compound interest earned in 3 years.

Here the principal amount; P = Rs.15625
Time; T = 1 year
i) Suppose the rate of interest be R% per annum.
So amount accumulated at the end of 1 year is given by;
$$\begin{gathered} \mathrm{A} = \mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{T}} = \mathrm{Rs}.16875 \\ \Rightarrow 15625{\left(1+\frac{\mathrm{R}}{100}\right)}^1 = 16875 \\ \Rightarrow 15625+\frac{15625\mathrm{R}}{100} = 16875 \\ \Rightarrow \frac{15625\mathrm{R}}{100} = 1250 \\ \Rightarrow \mathrm{R} = \frac{1250\times 100}{15625} = 8\% \end{gathered}$$
Therefore rate of interest is 8% per annum.
ii) Also the amount accumulated at the end of the second year (taking T = 2 years) is given by;
$$\begin{gathered} A_2 = P{\left(1+\frac{R}{100}\right)}^T \\ = 15625{\left(1+\frac{8}{100}\right)}^2 \\ = 15625\times {\left(\frac{27}{25}\right)}^2 \\ = 15625\times \frac{27}{25}\times \frac{27}{25} \\ = 18225 \end{gathered}$$
Therefore interest earned at the end of the second year = Rs.18225 - Rs.15625 = Rs.2600
iii) Now the amount accumulated at the end of third year (taking T = 3 years) is given by;
$$\begin{gathered} A_3 = P{\left(1+\frac{R}{100}\right)}^T \\ = 15625{\left(1+\frac{8}{100}\right)}^3 \\ = 15625\times {\left(\frac{27}{25}\right)}^3 \\ = 15625\times \frac{27}{25}\times \frac{27}{25}\times \frac{27}{25} \\ = 19683 \end{gathered}$$
Therefore amount accumulated at the end of third year is Rs.19683