Question

Three numbers whose sum is $15$ are in $A.P.$ If $1,4,19$ be added to them respectively, then they are, in $G.P.$ Find the numbers.

Answer 1

Let the three numbers in AP be,

$a-d,a,a+d$

Their sum => $3a=15$

=> $a=5$

Therefore numbers are, $5-d,5,5+d$

On adding $1,4,9$ respectively to these number, they become,

$6-d,9,24+d$

And these are GP (given)

So,

$96-d\left)\right(24+d)=92$

=> $d^2+18d-63=0$

=>$d=-21$ or $3$

Hence the numbers are,

$26,5,-16$ or $2,5,8$