A) 1
B) \[\frac{2}{\sqrt{5}}\]
C) \[\frac{\sqrt{5}-1}{2}\]
D) \[\frac{\sqrt{5}+1}{2}\]
Correct Answer: D
Solution :
Let first term and common ratio of G.P. are respectively \[a\] and \[r\], then under condition, \[{{T}_{n}}={{T}_{n-1}}+{{T}_{n-2}}\]\[\Rightarrow \]\[a{{r}^{n-1}}=a{{r}^{n-2}}+a{{r}^{n-3}}\] \[\Rightarrow \]\[a{{r}^{n-1}}=a{{r}^{n-1}}{{r}^{-1}}+a{{r}^{n-1}}{{r}^{-2}}\] \[\Rightarrow \]\[1=\frac{1}{r}+\frac{1}{{{r}^{2}}}\]\[\Rightarrow \]\[{{r}^{2}}-r-1=0\] \[\Rightarrow \]\[r=\frac{1\pm \sqrt{1+4}}{2}=\frac{1+\sqrt{5}}{2}\] Taking only (+) sign \[(\because \ r>1)\].
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