1. **State the problem:** We have a geometric sequence: 20, 100, 500, 2500, ... We want to find: a) The value of $h$ in the formula $h \times 5^{n-1}$ for the $n^{th}$ term. b) The value of $k$ in the formula $k \times 5^n$ for the $n^{th}$ term. 2. **Recall the formula for the $n^{th}$ term of a geometric sequence:** $$a_n = a_1 \times r^{n-1}$$ where $a_1$ is the first term and $r$ is the common ratio. 3. **Find the common ratio $r$:** $$r = \frac{100}{20} = 5$$ 4. **Find $h$ in $h \times 5^{n-1}$:** Since $a_n = a_1 \times r^{n-1}$ and $r=5$, we have $$a_n = 20 \times 5^{n-1}$$ So, $h = 20$. 5. **Find $k$ in $k \times 5^n$:** Rewrite $a_n = h \times 5^{n-1}$ as $$a_n = h \times 5^{n-1} = h \times \frac{5^n}{5} = \frac{h}{5} \times 5^n$$ Substitute $h=20$: $$a_n = \frac{20}{5} \times 5^n = 4 \times 5^n$$ So, $k = 4$. **Final answers:** - $h = 20$ - $k = 4$