Question
In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals
A.
$$\sqrt 5 $$
B.
$$\frac{1}{2}\left( {\sqrt 5 - 1} \right)$$
C.
$$\frac{1}{2}\left( {1 - \sqrt 5 } \right)$$
D.
$$\frac{1}{2}\sqrt 5 $$
Answer :
$$\frac{1}{2}\left( {\sqrt 5 - 1} \right)$$
Solution :
$$\eqalign{
& {\text{Let the series }}a,ar,a{r^2},.....{\text{ are in geometric progression}}{\text{.}} \cr
& {\text{given, }}a = ar + a{r^2} \cr
& \Rightarrow \,\,1 = r + {r^2} \cr
& \Rightarrow \,\,{r^2} + r - 1 = 0 \cr
& \Rightarrow \,\,r = \frac{{ - 1 \pm \sqrt {1 - 4 \times - 1} }}{2} \cr
& \Rightarrow \,\,r = \frac{{ - 1 \pm \sqrt 5 }}{2} \cr
& \Rightarrow \,\,r = \frac{{\sqrt 5 - 1}}{2}\left[ {\because {\text{ terms of G}}{\text{.P}}{\text{. are positive}}} \right. \cr
& \therefore r {\text{ should be }}\left. {{\text{positive}}} \right] \cr} $$