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# 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. $\mathrm{then}{r}^{n-1}={r}^{n}+{r}^{n+1}\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}1=r+{r}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}{r}^{2}+r-1=0\phantom{\rule{0ex}{0ex}}\therefore r=\frac{-1±\sqrt{5}}{2}$ Since r > 0 (given, g.p is positive series) $\mathrm{Hence}r=\frac{\sqrt{5}-1}{2}\phantom{\rule{0ex}{0ex}}\mathrm{Since}\mathrm{sin}18°=\frac{\sqrt{5}-1}{4}$ Therefore $r=2\left(\frac{\sqrt{5}-1}{4}\right)=2\mathrm{sin}18°$ Hence, the correct answer is option D.

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