Question: If the first 5 terms of an infinite geometric sequence are $100$, $60$, $36$, $21.6$, and $12.96$, then the sum of all the terms in the sequence is _______.
1. **State the problem:** We are given the first 5 terms of an infinite geometric sequence: $100$, $60$, $36$, $21.6$, and $12.96$. We need to find the sum of all terms in this infinite sequence.
2. **Recall the formula for the sum of an infinite geometric series:**
$$S_\infty = \frac{a}{1 - r}$$
where $a$ is the first term and $r$ is the common ratio, with the condition $|r| < 1$.
3. **Find the common ratio $r$:**
Calculate $r$ by dividing the second term by the first term:
$$r = \frac{60}{100} = 0.6$$
Check with other terms:
$$\frac{36}{60} = 0.6, \quad \frac{21.6}{36} = 0.6, \quad \frac{12.96}{21.6} = 0.6$$
So, $r = 0.6$.
4. **Calculate the sum of the infinite series:**
$$S_\infty = \frac{100}{1 - 0.6} = \frac{100}{0.4}$$
Show intermediate cancellation:
$$S_\infty = \frac{100}{\cancel{0.4}} = \frac{100}{\cancel{0.4}}$$
Simplify:
$$S_\infty = 250$$
5. **Conclusion:** The sum of all terms in the infinite geometric sequence is $250$.
**Answer: d.) 250**