The sum of three numbers in A.P. is 21. If the second number is reduced by 1 while third is increased by 1, three resulting numbers form an increasing G.P., the common ratio of the G.P. is

Let three numbers in A.P. be

$a – d, a, a + d.$

As $21 = (a – d) + a + (a + d) = 3a$

$\Rightarrow a=7$

Also, $7 – d, 7–1, 7+d+1$

.$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{6}^2=(7-d)(8+d)\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}36=56-d-d^2\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{d}^2+d-20=0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(d+5)(d-4)=0\Rightarrow d=-5,4\end{array}$

As, $7 – d, 6, 8 + d$ is an increasing G.P., $d = 4.$ The

common ratio of the G.P. is 

$\frac{6}{7-d}=\frac{6}{7-4}=2$