Question

# Let $$a, b, c$$ be positive integers such that $$\displaystyle \frac{b}{a}$$ 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 $$\displaystyle \dfrac{a^2+a-14}{a+1}$$ is ................

A
3
B
4
C
5
D
6

Solution

## The correct option is B $$4$$$$\frac{b}{a} = \frac{c}{b} = (integer)$$$$b^2 = ac \Rightarrow c = \frac{b^2}{a}$$$$\frac{a+b+c}{3} = b +2$$$$a+b+c = 3b + 6 \Rightarrow a - 2b + c= 6$$$$a-2b + \frac{b^2}{a} = 6 \Rightarrow 1 - \frac{2b}{a} + \frac{b^2}{a^2} = \frac{6}{a}$$$$\displaystyle \left ( \frac{b}{a}- 1 \right )^2 = \frac{6}{a} \Rightarrow a = 6 \ only$$Mathematics

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