# Compound Interest | Series and Sequences

Compound Interest is being used to calculate the total investment across over time. Suppose John invests \$1000 in the bank. He leaves the money in the bank for 4 years, and are paid an interest rate of 10% per annum. The interest is added to his investment each year, so the total value increases.

The percentage increase each year is 10%, so at the end of the year John will have $100\% + 10\% = 110\%$ of the value at its start. This corresponds to a multiplier of 1.1.

After one year the investment is worth;
$1000 \times 1.1 = 1100$
After two years the investment is worth;
$1000 \times 1.1^{2} = 1210$
After three years the investment is worth;
$1000 \times 1.1^{3} = 1331$
After four years the investment is worth;
$1000 \times 1.1^{4} = 1464.10$

## Practice Questions

### Question 1

$1000$ is invested for 5 years at 12% per annum compound interest, compounded annually. What will the amount to at the end of this period?
\begin{aligned} \displaystyle u_5 &= 5000 \times 1.12^{5} \\ &= 8811.71 \\ \end{aligned} \\

### Question 2

$1000$ is invested for 5 years at 12% per annum compound interest, compounded monthly. What will the amount to at the end of this period?
\begin{aligned} \displaystyle u_5 &= 5000 \times 1.01^{5 \times 12} \\ &= 9083.48 \\ \end{aligned} \\

### Question 3

How much should I invest now if the maturing value of $10 \ 000$ in 5 years’ time? The investment is at 12% per annum compounded quarterly.
\begin{aligned} \displaystyle u_1 \times 1.03^{5 \times 4} &= 10 \ 000 \\ u_1 &= \dfrac{10 \ 000}{1.03^{5 \times 4}} \\ &= 5536.76 \\ \end{aligned} \\
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