JEE Main 2016 (Offline)Question 1
If the $2^{nd}$, $5^{th}$ and $9^{th}$ terms of a non-constant arithmetic progression are in geometric progression, then the common ratio of this geometric progression is
Sequence and Series · Mixed Progressions (AP-GP)
Solution
Formulas used. (1) $a_n=a+(n-1)d$(2) $x,y,z$ in G.P. $\Rightarrow y^2=xz$Step 1 - G.P. condition. $$(a+4d)^2=(a+d)(a+8d)\ \Longrightarrow\ 16d^2+8ad=9ad+8d^2\ \Longrightarrow\ 8d^2=ad .$$ $d\ne0$, so $a=8d$. Step 2 - The three terms. $$a_2=9d,\qquad a_5=12d,\qquad a_9=16d .$$ Step 3 - Common ratio. $$\frac{12d}{9d}=\frac43 .$$ Answer: (D)