If the 2nd, 5th and 9th terms of a non-constant A.P.

Question

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

Accepted Answer

Let the A.P. have first term $A$ and common difference $d$: $a_2=A+d,\ a_5=A+4d,\ a_9=A+8d$.

G.P. condition: $(A+4d)^2=(A+d)(A+8d)$.

$$A^2+8Ad+16d^2=A^2+9Ad+8d^2 \Rightarrow 8d^2=Ad \Rightarrow A=8d\ (d\ne0).$$

So $a_2=9d,\ a_5=12d$, and $r=\dfrac{a_5}{a_2}=\dfrac{12d}{9d}=\dfrac43$.

Solutions · 1