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8. Sequences and Series
medium
If every term of a $G.P.$ with positive terms is the sum of its two previous terms, then the common ratio of the series is
A
$1$
B
$\frac{2}{{\sqrt 5 }}$
C
$\frac{{\sqrt 5 - 1}}{2}$
D
$\frac{{\sqrt 5 + 1}}{2}$
Solution
(d) 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 . $(\because \;r > 1)$
Standard 11
Mathematics
