Simple interest produces an arithmetic sequence because the same amount of interest is added in every time period. Find this fixed amount, use it as the common difference, and apply the arithmetic sequence formula.
For principal , annual interest rate written as a decimal, and time years, the interest added each year is
This remains constant because simple interest is calculated only on the original principal, not on the growing balance. This means the yearly interest amounts are equal and the balances increase linearly rather than exponentially.
If represents the balance at time , the balance after years is term . Therefore,
The total interest earned after years is , while the account balance is . Keep these quantities separate: a question asking for interest earned does not ask for the final balance.
A student invests \text{\}20004%$ simple interest per year. The annual interest is
The balances form the arithmetic sequence
After five years,
Equivalently,
| Feature | Simple interest | Compound interest |
|---|---|---|
| Interest calculation | Based only on the original principal | Based on the current balance |
| Sequence type | Arithmetic | Geometric |
| Change each period | Constant difference (Pr) | Constant ratio |
A common misconception is to use . That is the compound-interest formula, so it is incorrect when the question specifies simple interest.
Two indexing conventions are valid. If the initial balance is , use . If the first end-of-year balance is , then the balance after years is . Both simplify to . Label your timeline clearly.
This is IB Mathematics: Analysis and Approaches, SL 1.2, and it may appear on Paper 1 or Paper 2. Show the principal, convert the percentage rate to a decimal, identify , and then substitute into . Pay close attention to indexing: if is the initial balance, the balance after years is , not .
