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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°

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Solution

Let us tn, tn+1 and tn+2 denote nth, (n + 1)th and (n + 2)th term of geometric progression respectively.
According to given condition,
tn = tn+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 rn-1=rn+rn+1i.e 1= r+r2i.e r2+r-1=0r=-1±52
Since r > 0 (given, g.p is positive series)

Hence r=5-12Since sin18°=5-14
Therefore r=25-14=2sin 18°
Hence, the correct answer is option D.

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