Question

The second, first and third term of an arithmetic progression form a geometric progression in that order. Which of the following can be the common ratio of the geometric progression?

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $2$
• D. $-2$

Answer 1

Answer: (D)

$-2$

First term of AP is $=a$
Second term $=a+d$
Third term $=a+2d$
So the first three terms of GP are $a,a+d,a+2d$
In a geometric progression, there is a common ratio. So the ratio of the second term to the first term is equal to the ratio of the third term to the second term. So
$\frac{a+d}{a}=\frac{a+2d}{a+d}$
$a^2+2ad+d^2=a^2+2ad$
$d^2=0$
$d=\sqrt{2}$
The common ratio of the geometric progression, r, is equal to$\frac{(a+d)}{a}$
$\therefore$ if $d=\sqrt{2}$
then$r=\frac{a+\sqrt{2}}{a}=\frac{a\sqrt{2}}{a}=\sqrt{2}$