Question

In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P.
(a) sin 18°
(b) 2 cos 18°
(c) cos 18°
(d) 2 sin 18°

Answer 1

Let us t_n, t_n+1 and t_n+2 denote n^th, (n + 1)^th and (n + 2)^th term of geometric progression respectively.
According to given condition,
t_n = t_n+1 + t_n+2
i.e ar^n–1 = arⁿ + arⁿ⁺¹
where, a denotes first term of g.p and r denote the common ratio.
$$\begin{gathered} \mathrm{then}{r}^{n-1}=r^n+r^{n+1} \\ \mathrm{i}.\mathrm{e}1=r+r^2 \\ \mathrm{i}.\mathrm{e}{r}^2+r-1=0 \\ \therefore r=\frac{-1\pm \sqrt{5}}{2} \end{gathered}$$
Since r > 0 (given, g.p is positive series)

$$\begin{gathered} \mathrm{Hence}r=\frac{\sqrt{5}-1}{2} \\ \mathrm{Since}\sin 18°=\frac{\sqrt{5}-1}{4} \end{gathered}$$
Therefore $r=2\left(\frac{\sqrt{5}-1}{4}\right)=2\sin 18°$
Hence, the correct answer is option D.