First slide

Geometric progression

Question

Let a, b and c from a G.P of common ratio r, with $0 < r < 1.$ If a, 2b and 3c form an A.P. then r equals ___________

Easy

Solution

$b = a r, c = a r^{2}$
$a, 2 b, 3 c are in A .P$
$\Rightarrow 4 b = a + 3 c$
$\Rightarrow 4 a r = a + 3 a r^{2}$
$\Rightarrow 3 r^{2} - 4 r + 1 = 0$
$\Rightarrow r = 1, \frac{1}{3} (where r \neq 1)$
$So, r = \frac{1}{3}$

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