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8. Sequences and Series
medium
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
$\frac{{\sqrt 5 - 1}}{2}$
B
$\frac{{1 - \sqrt 5 }}{2}$
C
$1$
D
$2\sqrt 5 $
Solution
(a) Under condition ${T_n} = {T_{n + 1}} + {T_{n + 2}}$
$ \Rightarrow a{r^{n – 1}} = a{r^n} + a{r^{n + 1}} $
$\Rightarrow {r^{n – 1}} = {r^n}(1 + r)$
$ \Rightarrow $ ${r^2} + r – 1 = 0$
$ \Rightarrow r = \frac{{ – 1 \pm \sqrt {1 + 4} }}{2} = \frac{{ – 1 \pm \sqrt 5 }}{2}$
Since, each term is $ + ve$.
Hence common ratio is $\frac{{\sqrt 5 – 1}}{2}$.
Standard 11
Mathematics