The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms.
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
Let the first term be a and the common difference be r.
$\therefore a_{1}+a_{2}=5$
$\Rightarrow a+a r=5$ ....(i)
Also, $a_{n}=3\left[a_{n+1}+a_{n+2}+a_{n+3}+\ldots \infty\right] \forall n \in N$
$\Rightarrow a r^{n-1}=3\left[a r^{n+1}+a r^{n+2}+a r^{n+3}+\ldots \infty\right]$
$\Rightarrow a r^{n-1}=\frac{3 a r^{n}}{1-r}$
$\Rightarrow 1-r=3 r$
$\Rightarrow 4 r=1$
$\Rightarrow r=\frac{1}{4}$
Putting $r=\frac{1}{4}$ in (i) :
$a+\frac{a}{4}=5$
$\Rightarrow 5 a=20$
$\Rightarrow a=4$
Thus, the G.P. is $4,1, \frac{1}{4}, \frac{1}{16}, \ldots \infty$.
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