Question

Find the numbers a,b,c between 2 and 18 such that (i) their sum is 25 (ii) the numbers 2,a,b are consecutive terms of an A.P. and (iii) the numbers b,c,18 are consecutive terms of a G.P.

Answer 1

We have given $a+b+c=25...\left(1\right)$
$2a=b+2...\left(2\right)$
and $c^2=18b$ ...(3)
Eliminating $a$ from (1) and (2),
$b=16-\frac{2c}{3}.$
Then from (3), $c^2=18(16-\frac{2c}{3})$
or $c^2+12c-18\times 16=0$
or $(c-12)(c+24)=0$
$c=-24$
is rejected since it does not lie between 2 and 18.
Hence $c=12.$ Then
(3) gives $b=8$ and finally. (2) gives $a=5.$
Thus $a=5,b=8$ and $c=12.$