In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals
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Official solution
Let the series a,ar,ar2,........ are in geometric progression.
given, a=ar+ar2
\[ \Rightarrow 1 = r + {r^2}\]
\[ \Rightarrow {r^2} + r - 1 = 0\]
\[ \Rightarrow r = {{ - 1 \mp \sqrt {1 - 4 \times - 1} } \over 2}\]
\[ \Rightarrow r = {{ - 1 \pm \sqrt 5 } \over 2}\]
\[ \Rightarrow r = {{\sqrt 5 - 1} \over 2}\]
[ As terms of G.P. are positive
\[\therefore\] r should be positive]