Question

If an employee deposits Rs. 2,000 at the end of each year into his company’s plan which pays 7% interest compounded quarterly, how much will he have in the account at the end of 5 years?

Answer 1

The formula for compound interest

$$A=P(1+{r \over n})^{nt}$$

A= the future value of the investment/loan, including interest

P = the principal investment amount

r = the annual interest rate (decimal)

n = the number of times that interest is compounded per unit t

t = the time the money is invested or borrowed for

We have that $P_0=2000, r=0.07, n=4$

At the end of the second year

$$A_2=P_0\Big(1+(1+{r \over n})^n\Big)$$

At the end of the third year

$$A_3=P_0\Big(1+(1+{r \over n})^n+\big((1+{r \over n})^n\big)^2\Big)$$

We see the geometric series

$$S_k=1\cdotp{1-\big((1+{r \over n})^n\big)^k \over 1-(1+{r \over n})^n}$$

Then

$$A_m=P_0S_m=P_0\cdotp{1-\big((1+{r \over n})^n\big)^m \over 1-(1+{r \over n})^n}$$

How much will he have in the account at the end of 5 years?

$$A_6=2000\cdotp{1-\big((1+{0.07 \over 4})^4\big)^6 \over 1-(1+{0.07 \over 4})^4}=14373.78$$

He will have in the account at the end of 5 years RM 14373.78.