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Sum of n Terms

Prove that a,...

Question

Prove that a,b,c are in A.P., G.P. or H.P. according as the value of $\frac{a - b}{b - c}$ is equal to $\frac{a}{a}, \frac{a}{b} o r \frac{a}{c}$ respectively

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Solution

$\frac{a - b}{b - c} = \frac{a}{a} g i v e s b = \frac{a + c}{2} ∴ i n A . P .$
$\frac{a - b}{b - c} = \frac{a}{b} g i v e s b^{2} = a c ∴ i n G . P .$
$\frac{a - b}{b - c} = \frac{a}{c} g i v e s b = \frac{2 a c}{a + c} ∴ i n H . P .$

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