Q37 / 90 · JEE Main 2016 (03 Apr) · Mathematics

If the $2^{n d}, 5^{t h}$ and $9^{t h}$ terms of a non-constant arithmetic progression are in geometric progression, then the common ratio of this geometric progression is

A
$1$
B
$\frac{7}{4}$
C
$\frac{8}{5}$
Correct answer
D
$\frac{4}{3}$

Solution

Let the A.P. be

a, a + d, a + 2 d, . . . . .

Given,

(a + d) . (a + 8 d) = {(a + 4 d)}^{2}

a^{2} + 9 a d + 8 d^{2} = a^{2} + 8 a d + 16 d^{2}

8 d^{2} - a d = 0

d [8 d - a] = 0

∵

d \neq 0

d = \frac{a}{8}

So,

2^{n d}

term,

5^{t h}

term,

9^{t h}

term

would be

(a + \frac{a}{8})

,

(a + \frac{a}{2})

,

(a + a)

\frac{9 a}{8}

,

\frac{3 a}{2}

,

2 a

Common Ratio

= \frac{2^{n d} t e r m}{1^{s t} t e r m} = \frac{3 a \times 8}{2 \times 9 a} = \frac{4}{3}

Hence, the correct answer is option: 'D'.

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