Question

In a geometrical progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals to

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

Answer 1

The correct option is B $\frac{\sqrt{5}-1}{2}$

Given that each term in the GP is equal to the sum of next two terms. Let $a$ be the first term and common ratio be $r$.

Then $a=ar+a{r}^2$

$a\ne 0\therefore 1=r+r^2$

$r^2+r-1=0$

$⟹r=\frac{-1\pm \sqrt{1+4}}{2}=\frac{-1\pm \sqrt{5}}{2}$

The GP has only positive terms. SO common ratio is positive.

Hence common ratio $=\frac{\sqrt{5}-1}{2}$