Question

If three non-zero numbers are in A.P. and the squares of those numbers taken in same order are in G.P., then the common ratio can be

A. 0
B. 1
C. 3+2√2
D. 3−2√2

Answer 1

If three non-zero numbers are in arithmetic progression (A.P.), and the squares of those numbers taken in the same order are in geometric progression (G.P.), then the common ratio of the G.P. is 0.

Step-by-step explanation:

Let the three numbers be a-d, a, and a+d, where 'a' is the common difference of the arithmetic progression (A.P.).

According to the problem, the squares of these numbers taken in the same order are in a geometric progression (G.P.). So, (a-d)^2, a^2, and (a+d)^2 are in G.P.

Using the property of G.P., we have (a^2) / ((a-d)^2) = ((a+d)^2) / (a^2). Cross-multiplying and simplifying, we get (a^4) - 2a^2d^2 + (d^4) = (a^4) + 2a^2d^2 + (d^4). Simplifying further, we get 4a^2d^2 = 0.

Since the three numbers are non-zero, a^2 and d^2 cannot be zero. Therefore, the common ratio of the G.P. is 0. Thus, the answer is (A) 0.