Find three numbers $a, b, c$ between 2 and 18 such that (i) their sum is 25 (ii) $2, a, b$ are conse

Question

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

Professor PC Thomas Classes · Answer

$a+b+c=25$ $$\begin{align*} & 2 a=b+2  	ag{2}\ & c^{2}=18 b 	ag{1} \end{align*}$$ $$\begin{equation*}\Rightarrow \frac{b+2}{2}+b+\sqrt{18 b}=25 	ag{3} \end{equation*}$$ $\Rightarrow 3 b+6 \sqrt{2 b}-48=0$\ $\Rightarrow \quad b+2 \sqrt{2 b}-16=0$\ $\Rightarrow \quad b+4 \sqrt{2} \sqrt{b}-2 \sqrt{2} \sqrt{b}-16=0$\ $\Rightarrow \quad(\sqrt{b}-2 \sqrt{2})(\sqrt{b}+4 \sqrt{2})=0$\ $\Rightarrow \quad b=8, a=5, c=12$.\