Question 0419

An arithmetic series has first term ${a}$ and common difference ${2a,}$ where ${a \neq 0.}$ A geometric series has first term ${a}$ and common ratio ${2.}$ The ${k\textrm{th}}$ term of the geometric series is equal to the sum of the first ${8}$ terms of the arithmetic series. Find the value of ${k.}$

The first term of an arithmetic series is non-zero. The sum of the first two terms of the series is ${2}$ and the product of the first two terms of the series is ${0.}$ Find the ${9\textrm{th}}$ term of the series.