Let the A.P. be $a, a+d, a+2d, \ldots$ Given, $(a+d)(a+8d) = (a+4d)^2$
Step-by-Step Solution
Key Concept: Expand the given quadratic condition and solve for the relationship between $a$ and $d$ in the arithmetic progression.
Expanding the given condition: $a^2 + 9ad + 8d^2 = a^2 + 8ad + 16d^2$. Simplifying: $8d^2 - ad = 0$, so $d(8d - a) = 0$. Since $d \neq 0$, we have $a = 8d$. The 2nd term is $a + d = 8d + d = 9d$, the 9th term is $a + 8d = 16d$, and terms are $9d, 10d, 11d, \ldots$ with common ratio $r = \frac{10d}{9d} = \frac{10}{9}$.
Correct Answer: 8