Question

Three non-zero numbers $a,b$ and $c$ are in A.P.. If increasing $a$ by $1$ or increasing $c$ by $2,$ then the numbers are in G.P., then the value of $b$ is

Answer 1

$a,b$ and $c$ are in A.P., so,
$2b=a+c\dots \left(1\right)$
Again, by the given condition, $a+1,b$ and $c$ are in G.P. and $a,b$ and $c+2$ are in G.P. Hence,
$b^2=(a+1)c\dots \left(2\right)$
and
$b^2=a(c+2)\dots \left(3\right)$
By (2) and (3),
$(a+1)c=a(c+2)$
$\Rightarrow ac+c=ac+2a$
$\Rightarrow c=2a$
Equation (2) gives $b^2=(a+1)2a$
Also, Eq. (1) gives
$2b=a+2a=3a$
$\Rightarrow b=\frac{3a}{2}$
$\Rightarrow \frac{9a^2}{4}=(a+1)2a$
$\Rightarrow \frac{9a}{8}=a+1$
$\Rightarrow a=8$
$\Rightarrow c=2\left(8\right)=16$
$\Rightarrow 2b=8+16=24$
$\Rightarrow b=12$