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Question: The terms of a G.P. are positive. If each term is equal to the sum of two terms that follow it, then...

The terms of a G.P. are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is.

A

512\frac { \sqrt { 5 } - 1 } { 2 }

B

152\frac { 1 - \sqrt { 5 } } { 2 }

C

1

D

252 \sqrt { 5 }

Answer

512\frac { \sqrt { 5 } - 1 } { 2 }

Explanation

Solution

Under condition Tn=Tn+1+Tn+2T _ { n } = T _ { n + 1 } + T _ { n + 2 }

arn1=arn+arn+1rn1=rn(1+r)\Rightarrow a r ^ { n - 1 } = a r ^ { n } + a r ^ { n + 1 } \Rightarrow r ^ { n - 1 } = r ^ { n } ( 1 + r )

\Rightarrow r2+r1=0r=1±1+42=1±52r ^ { 2 } + r - 1 = 0 \Rightarrow r = \frac { - 1 \pm \sqrt { 1 + 4 } } { 2 } = \frac { - 1 \pm \sqrt { 5 } } { 2 }

Since, each term is +ve+ v e.

Hence common ratio is 512\frac { \sqrt { 5 } - 1 } { 2 } .