3.2 Compound Interest using Sequences

Let P is the principal sum and i the interest rate per annum, mathematically, we can calculate the total balance at the ends of successive years for n years as:

At the end of the 1^st year:

The interest due              = Pi, then the future value = P + Pi = P(1+i)

At the end of the 2^nd year:

The interest due              = [P(1+i)]i, then the future value = P (1+i) + [P(1+i)]i

= P(1+i) (1+i)

= P(1+i)²

At the end of the 3^rd year:

The interest due              = [P(1+i)²]i, then the future value = P (1+i)² + [P(1+i)²]i

= P(1+i)²(1+i)

= P(1+i)³

and so on.

Here, we can see that the total balance or the future balance value at the end of n years forms a geometric progression with first term A and common ratio (1+r). Thus, to obtain the total balance at the end of n years, we can use the above formula for the nth term of a geometric progression with n-1 replaced by n as follows:

S = P(1+i)ⁿ

Here, the total balance at the beginning of the nth year is given by:

S = P(1+i)^n-1

However, to note that the above calculation is express as rates per annum or called as nominal rates. In some cases, the interest period need not be a year and for example, interest is “payable half-yearly’ or ‘compounded half-yearly’, interest period is payable half a year. Then, the interest rate need to be divided by two to give the effective rate per half year. The number of times you pay interest or compounding on your principal is important to calculate the total balance or future value.