Simple interest is modelled by an arithmetic sequence because the same fixed amount of interest is added during every time period. The account balance therefore increases by a constant common difference.
For an initial principal and an interest rate per period, written as a decimal, the interest added each period is
After periods, the balance is
This form uses , so the initial investment is term zero. If the sequence begins with the balance after the first interest payment, then
and the standard arithmetic-sequence formula gives
For example, suppose \text{\}10005%$ simple interest per year. The annual interest is
The balances are therefore . After years,
| Simple interest | Compound interest |
|---|---|
| Adds the fixed amount (Pr) each period | Adds a changing amount because interest is calculated on the new balance |
| Produces an arithmetic sequence | Produces a geometric sequence |
| Constant common difference | Constant common ratio |
The sequence can also be recognized from its first differences. Subtracting consecutive balances gives the same value each time, confirming an arithmetic model. In the example, each subtraction gives fifty dollars. The gradient of a graph of balance against time is therefore fifty dollars per year, while the vertical intercept is the original principal. This linear interpretation is useful when a question provides a table or graph instead of stating the interest method directly in the context explicitly.
A common misconception is that every interest problem uses a geometric sequence. This is incorrect: simple interest is arithmetic, while compound interest is geometric.
This is SL 1.2, Sequences and Sigma Notation, studied at both SL and HL. In a Paper 1 short-response or Paper 2 contextual question, identify , calculate , state the arithmetic model, and check whether the initial amount is labelled term or term . Keep full GDC precision and give money to two decimal places when required.