Question

if 3rd, 6th and 11th terms of an arithmetic progression are in geometric progression, then the common ratio of geo metric progression is​

Answer 1

for the arithmetic progression,

let the first term be a and common difference be d

so the terms are a, a+d, a+2d.. a+5d...

2nd term is a+d

3rd term is a+2d

5th term is a+5d

now if these are in GP,

(a+2d) / (a+d) = (a+5d) / (a+2d)

(a+2d)^2 = (a+d)(a+5d)

a^2 + 4ad + 4d^2 = a^2 + 6ad + 5d^2

d^2 = -2ad (Assuming d is not equal to 0 else we wont have an AP initially)

or d = -2a ... (1)

Now, using this to determine the ratio

(a+2d) / (a+d) = (a-4a) / (a-2a) = -3a/-a = 3

So,

common ratio is 3.

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