1. **State the problem:** We are given the first four terms of a geometric progression (GP): 1250, 250, 50, 10. We need to find: a) The common ratio $r$ of the progression. b) The fifth term of the progression. 2. **Recall the formula for a geometric progression:** The $n$-th term of a GP is given by: $$a_n = a_1 \times r^{n-1}$$ where $a_1$ is the first term and $r$ is the common ratio. 3. **Find the common ratio $r$:** The common ratio is the factor by which we multiply one term to get the next term. Calculate $r$ by dividing the second term by the first term: $$r = \frac{a_2}{a_1} = \frac{250}{1250}$$ Simplify the fraction: $$r = \frac{\cancel{250}}{\cancel{1250}} = \frac{1}{5}$$ So, the common ratio is $r = \frac{1}{5}$. 4. **Find the fifth term $a_5$:** Use the formula: $$a_5 = a_1 \times r^{4} = 1250 \times \left(\frac{1}{5}\right)^4$$ Calculate $\left(\frac{1}{5}\right)^4$: $$\left(\frac{1}{5}\right)^4 = \frac{1^4}{5^4} = \frac{1}{625}$$ Multiply: $$a_5 = 1250 \times \frac{1}{625}$$ Simplify: $$a_5 = \frac{\cancel{1250}}{\cancel{625}} \times 1 = 2$$ **Final answers:** a) The common ratio is $\frac{1}{5}$. b) The fifth term is $2$.