Financial Mathematics • Topic 3 of 6

Growth Models

Many quantities grow by a fixed percentage each period — populations, audiences, investments. The model is the same as compound interest: A = P(1 + r)^t, where P is the starting amount, r the growth rate as a decimal, and t the number of periods. A base greater than 1 (like 1.2 for 20% growth) signals growth. For a quantity of 500 growing 20% per period, after 2 periods it is 500(1.2)² = 720. Read the rate carefully and raise (1 + r) to the number of periods. Recognising whether a written model represents growth or decay from its base is a frequent SAT question.

Amount versus time: a single blue convex curve rising more steeply over time, showing exponential growthGrowthTimeAmountPA = P(1 + r) to the power t

✅ Solved examples

1. 500 grows 20% per period. Value after 2 periods?
500(1.2)² = 500 × 1.44 = 720.
2. 200 grows 10% per period. Value after 2 periods?
200(1.1)² = 200 × 1.21 = 242.
3. Does y = 300(1.5)^t show growth or decay?
Base 1.5 > 1, so growth.
4. 100 grows 50% per period. Value after 2 periods?
100(1.5)² = 100 × 2.25 = 225.

✏️ Practice — try these, take hints as needed

1. 400 grows 10% per period. Value after 2 periods?
A = P(1 + r)².
400 × 1.21.
—
484.
2. 600 grows 20% per period. Value after 2 periods?
600 × 1.44.
—
—
864.
3. Does y = 250(1.1)^t represent growth or decay?
Look at the base.
1.1 > 1.
—
Growth.
4. 200 grows 50% per period. Value after 2 periods?
200 × 2.25.
—
—
450.
5. 1,000 grows 10% per period. Value after 2 periods?
1000 × 1.21.
—
—
1,210.

📝 Topic test — 8 questions

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