Question

In a geometric progression consisting of positive terms, each terms equals the sum of the next two terms. Then the common ratio of this progression equals:

• A. $\frac{1}{2}(1-\sqrt{5})$
• B. $\frac{1}{2}\sqrt{5}$
• C. $\sqrt{5}$
• D. $\frac{1}{2}(\sqrt{5}-1)$

Answer 1

The correct option is D $\frac{1}{2}(\sqrt{5}-1)$

Let the geometric progression consisting of positive terms be $\alpha ,\alpha \pi ,alpha{\pi }^2$.

Since each term equals sum of next two terms.

$\therefore \alpha =\alpha \pi +{\pi }^2$

$\alpha =\alpha (\pi +{\pi }^2)$

$\therefore \pi +{\pi }^2=1$

$\therefore \pi =\frac{-1\pm \sqrt{1+4}}{2}$

$=\frac{-1\pm \sqrt{5}}{2}$

Since, the terms of G.P are positive

$\therefore$ common ration $\pi =\frac{-1\pm \sqrt{5}}{2}$

$=\frac{1}{2}(\sqrt{5}-1)$