Question

the second,third and sixth term of an ap are consecutive terms of a gp.find common ratio​

Answer 1

Answer:

The common ratio is 3

Step-by-step explanation:

Let$t_1,t_2,.........$ be the given A.P

Given:

$t_2,t_3,t_6$ are in G.P

$\implies\:\frac{t_3}{t_2}=\frac{t_6}{t_3}=r$

$\implies\:\frac{a+2d}{a+d}=\frac{a+5d}{a+2d}=r$.......(1)

$\implies\:(a+2d)^2=(a+d)(a+5d)$

$\implies\:a^2+4d^2+4ad=a^2+6ad+5d^2$

$\implies\:4d^2-5d^2=6ad-4ad$

$\implies\:-d^2=2ad$

$\implies\:d=-2a$

put d=-2a in (1), we get

$\implies\:r=\frac{a+2(-2a)}{a+(-2a)}$

$\implies\:r=\frac{a-4a}{a-2a}$

$\implies\:r=\frac{-3a}{-a}$

$\implies\:r=3$