Question

In a G.P. series consisting of positive terms, each term is equal to the sum of next two terms. Then the common ratio of this G.P. series is

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

Answer 1

Answer: (B)

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

Each term is sum of next two terms

$⟹t_n=t_{n+1}+t_{n+2}$

$⟹a{r}^{n-1}=a{r}^n+a{r}^{n+1}$

$⟹1=r+r^2$

$⟹r^2+r-1=0$

$⟹r=\frac{-1\pm \sqrt{1-4(-1)}}{2\left(1\right)}$

$⟹r=\frac{-1\pm \sqrt{5}}{2}$

$⟹r=\frac{\sqrt{5}-1}{2}$ or $r=\frac{-\sqrt{5}-1}{2}$

But common ration can' be negative, as all the terms are positive.

$\therefore r=\frac{\sqrt{5}-1}{2}$