1. **Problem statement:** The number of cases of a disease increases by the same factor each year. Given the initial number of cases and the number of cases at the end of several years, find an expression for the number of cases after $n$ years. 2. **Identify the initial value and growth factor:** - Initial number of cases (at start, year 0): $1400$ - Number of cases at the end of year 1: $2100$ - Number of cases at the end of year 2: $3150$ - Number of cases at the end of year 3: $4725$ 3. **Find the growth factor $r$:** The number of cases grows by the same factor each year, so $$2100 = 1400 \times r$$ Divide both sides by 1400: $$\frac{2100}{1400} = \cancel{1400} \times r \div \cancel{1400}$$ $$1.5 = r$$ 4. **Verify the growth factor with other years:** End of year 2: $$3150 = 2100 \times r = 2100 \times 1.5$$ $$3150 = 3150$$ (correct) End of year 3: $$4725 = 3150 \times r = 3150 \times 1.5$$ $$4725 = 4725$$ (correct) 5. **General expression:** The number of cases after $n$ years is given by the formula for exponential growth: $$\text{Number of cases after } n \text{ years} = 1400 \times 1.5^n$$ This means you multiply the initial number of cases by $1.5$ raised to the power of $n$ to find the number of cases after $n$ years. **Final answer:** $$\boxed{1400 \times 1.5^n}$$