Arithmetic and geometric sequences – simple and compound interest

What it shows

An arithmetic sequence adds the same common difference d at every step, so uₙ = u₁ + (n − 1)d is a linear function of n and the points (n, uₙ) lie on a straight line. A geometric sequence multiplies by the same common ratio q, so uₙ = u₁·qⁿ⁻¹ and the points follow an exponential curve: growth when q > 1, decay towards 0 when 0 < q < 1, alternating signs when q < 0. Simple interest adds a fixed amount each period and forms an arithmetic sequence; compound interest multiplies by 1 + r each period and forms a geometric sequence.

How to use

Click Arithmetic, set First term u₁ and Common difference d, and check that the points stay on a line even when d is negative. Click Geometric and move Common ratio q above 1, between 0 and 1, and below 0. Click a bar or drag the k slider to read uₖ and Sₖ from the formulas. In Simple/compound interest mode, change Interest rate per period and Number of terms n and watch the gap between the two bars widen.

Parameters you can change

• Sequence type Arithmetic sequence, Geometric sequence, Simple and compound interest
• First term u₁ (initial principal in interest mode) -20–100
• Common difference d (arithmetic sequence) -10–10
• Common ratio q (geometric sequence) -2–3
• Number of terms n (number of periods) 3–30
• Interest rate per period (interest mode) 1–30 %
• Currency From the language, US dollar, Euro, Indian rupee, Indonesian rupiah, Saudi riyal

Questions to explore

1. Why do the points of an arithmetic sequence always lie on a straight line?
2. With q = −1, how do the terms and the sums Sₙ behave?
3. At 8% per period, after how many periods does compound interest exceed simple interest by more than 10% of the principal?