In a G.P. of positive terms, if any term is equal to the sum of the next two terms.

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°

Solution:

Let us tntn+1 and tn+2 denote nth, (n + 1)th and (n + 2)th term of geometric progression respectively.

According to given condition,

ttn+1 + tn+2

i.e arn–1 = arn arn+1

where, a denotes first term of g.p and r denote the common ratio.

then $r^{n-1}=r^{n}+r^{n+1}$

i. e $1=r+r^{2}$

i. e $r^{2}+r-1=0$

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

Since r > 0 (given, g.p is positive series)

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

Since $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$

Therefore $r=2\left(\frac{\sqrt{5}-1}{4}\right)=2 \sin 18^{\circ}$

Hence, the correct answer is option D.

 

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