Question

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then common ratio of the G.P. is

• A. $\frac{1}{2}(1-\sqrt5)$
• B. $\frac{1}{2}\sqrt5$
• C. $\sqrt5$
• D. $\frac{1}{2}(\sqrt5-1)$

Answer 1

Answer: (D)

$\frac{1}{2}(\sqrt5-1)$

Answer 2

Let $a$ be the first term and $r$ be the common ratio of the $G.P$. Now $T_{1}= T_{2}+T_{3} \quad$ (As given) $\Rightarrow a=ar+ar^{2}$ $\Rightarrow r^{2}+r-1 =0$ $\Rightarrow r= \frac{-1\pm\sqrt{1+4}}{2} = \frac{-1\pm\sqrt{5}}{2}$ Since $G.P$. contains positive terms $\therefore r>o$ $\therefore r =\frac{ \sqrt{5}-1}{2}$