For a- b- c to be in GP- what is the value of a-b-b-c

For a, b, c to be in GP, we must have b/a = c/b=r or b/a = c/b = 1/r Case I When b/a = c/b = r In this case, b = ar and c = br. ∴ a b/b c = a ar/b br = a ...

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Byju's Answer

Standard XI

Mathematics

Selecting Consecutive Terms in GP

For a, b, c t...

Question

For a, b, c to be in GP, what is the value of $\frac{a - b}{b - c}$

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Solution

For a, b, c to be in GP, we must have

$\frac{b}{a} = \frac{c}{b} = r$ or $\frac{b}{a} = \frac{c}{b} = \frac{1}{r}$

Case I When $\frac{b}{a} = \frac{c}{b} = r$

In this case, b = ar and c = br.

$∴ \frac{a - b}{b - c} = \frac{a - a r}{b - b r} = \frac{a (1 - r)}{b (1 - r)} = \frac{a}{b}$

Case II When $\frac{b}{a} = \frac{c}{b} = \frac{1}{r}$

In this case, a = br and b = cr.

$∴ \frac{a - b}{b - c} = \frac{b r - b}{c r - c} = \frac{b (r - 1)}{c (r - 1)} = \frac{b}{c}$

Hence, the value of $(\frac{a - b}{b - c})$ is $\frac{a}{b}$ or $\frac{b}{c}$ .

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For a, b, c to be in GP, what is the value of a−bb−c

For a, b, c to be in GP, what is the value of $\frac{a - b}{b - c}$