Question

Three non zero numbers $a,b,$ and $c$ are in A.P. If we increase $a$ by $1$ or increase $c$ by $2,$ the numbers become in G.P. Find the value of $b-a$.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D $4$

Given: $a,b,c$ are in A. P

$\Rightarrow 2b=a+c$

And given $a+1,b,c$ are in G.P

$\Rightarrow b^2=(a+1)\left(c\right)$

$\Rightarrow b^2=(a+1)(2b-a)$

$\Rightarrow b^2=2ab-a^2+2b-a$

$\Rightarrow a^2-2ab+b^2=2b-a$

$\Rightarrow (a-b{)}^2=2b-a$

And $a,b,c+2$ are in GP

$\Rightarrow b^2=a(c+2)$

$\Rightarrow b^2=a(2b-a+2)$

$\Rightarrow b^2=2ab-a^2+2a$

$\Rightarrow (a-b{)}^2=2a$

$\Rightarrow 2b-a=2a$

$\Rightarrow 2b=3a$

$\Rightarrow {(a-\frac{3a}{2})}^2=2a$

$\Rightarrow \frac{a^2}{4}=2a$

$\Rightarrow a=8$

$\Rightarrow b=12$

$\Rightarrow b-a=4$