The fourth, seventh and sixteenth terms of an arithmetic sequence also form consecutive terms of a geometric sequence.

Question

Find the common ratio of the geometric sequence.

Now how do you display the consecutive terms of a geometric sequence as you don't know what the power of $r$ is, or do you use like $r^{(n+1)}$ , $r^{(n+2)}$, so on…

Answer 1

If the arithmetic sequence has first term $a$ and common difference $d$ then its fourth term is $a+3d$, its seventh term is $a+6d$ and its sixteenth term is $a+15d$. If these three terms form a geometric sequence with common ratio $r$ then

$r = \frac{a+6d}{a+3d}=\frac{a+15d}{a+6d} \\ \Rightarrow (a+6d)^2 = (a+3d)(a+15d) \\ \Rightarrow a^2 +12ad + 36d^2 = a^2 +18ad + 45d^2$

I'll let you take it from there.