Question

If the three numbers $a,b,c$ which are between $2$ and $18$ and also satisfy the below conditions :
(i) their sum is $25$
(ii) the numbers $2,a,b$ are consecutive terms of an AP
(iii) the numbers $b,c,18$ are consecutive terms of a GP.
Find $c-b+a$?

Answer 1

Given $a+b+c=25$
$2,a,b$ in $AP$$\Rightarrow 2+b=2a$
$b,c,18$ in $GP$$\Rightarrow c^2=18b$
From above 3 equations we have
$3a-27=-c$ $\Rightarrow 9(a^2-18a+81)=c^2$
$\Rightarrow 9(a^2-18a+81)=18(2a-2)$
$\Rightarrow a^2-22a+85=0$
$\Rightarrow (a-5)(a-17)=0$
$\Rightarrow a=5$ or $a=17$ if we take $a=17$ $b,c$ don't lie in $2$ and $18$ so $a=5\Rightarrow b=8\Rightarrow c=12$
$\Rightarrow c-b+a=9$