Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geomet

Question

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2$, then the value of $\frac{a^{2}+a-14}{a+1}$ is

Professor PC Thomas Classes · Answer

(4) : $\frac{b}{a}=\frac{c}{b}=$ integer (say)\ $\frac{a+b+c}{3}=b+2$\ (Given)\ $\Rightarrow a+b+c=3 b+6 \Rightarrow a-2 b+c=6$\ $\Rightarrow a-2 b+\frac{b^{2}}{a}=6 \Rightarrow \frac{b^{2}}{a^{2}}-\frac{2 b}{a}+1=\frac{6}{a}$\ $	herefore\left(\frac{b}{a}-1ight)^{2}=\frac{6}{a}=$ perfect square.\ $	herefore \quad a=6$ only\ Thus, $\frac{a^{2}+a-14}{a+1}=\frac{36+6-14}{7}=\frac{28}{7}=4$\