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

Engineering

Mathematics

Arithmetic Mean Geometric Mean and Harmonic Mean

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 :

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a, ar, ar² where a, r $\in$ N

Now, $\frac{a + b + c}{3} = b + 2$ ⇒ a(r – 1)2 = 6 $\Rightarrow {(r - 1)}^{2} = \frac{6}{a}$

$∴$ a = 6 and r = 2

So, $\frac{a^{2} + a - 14}{a + 1} = 4$