1. Tyler has an infinite geometric series with sum 10 . He increases the first term of his sequence by 4 and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum 15. Compute the common ratio of Tyler's series.
Proposed by: Isabella Quan
Answer: $\square$
Solution: Let $a$ and $r$ be the first term and common ratio of the original series, respectively. Then $\frac{a}{1-r}=10$ and $\frac{a+4}{1-r}=15$. Dividing these equations, we get that
$$
\frac{a+4}{a}=\frac{15}{10} \Longrightarrow a=8 \text {. }
$$

Solving for $r$ with $\frac{a}{1-r}=\frac{8}{1-r}=10$ gives $r=\frac{1}{5}$.