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Home  >>  CBSE XI  >>  Math  >>  Sequences and Series
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The first term of a G.P. is $1$. The sum of third term and fifth term is $90$. Find the common ratio of the G.P.

$\begin{array}{1 1}\pm 2 \\ \pm 3 \\ \pm 9 \\ \pm 6 \end{array} $

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  • $n^{th}$ term of a G.P. $=t_n=a.r^{n-1}$
Given: $1^{st}$ term =1 and sum of $3^{rd}$ and $5^{th}$$= 90$
$\Rightarrow\:a=1\:\: and \:\: t_3+t_5=90$
We know that $n^{th}$ term of a G.P. is $t_n=a.r^{n-1}$
$\Rightarrow\:t_3=a.r^{2-1}=1\times r^2=r^2$ and
$t_5=a.r^{5-1}=1\times r^4=r^4$.
$\therefore\:r^2+r^4=90$
Let $r^2=x$
then $x+x^2=90$
$\Rightarrow\:x^2+x-90=0$
$\Rightarrow\:x^2+10x-9x-90=0$
$\Rightarrow\:x(x+10)-9(x+10)=0$
$\Rightarrow\:(x-9)(x+10)=0$
$\Rightarrow\:x=-10\:\:or\:\:x=9$
But since $r^2$ cannot be negative, $x=r^2=9$
$\Rightarrow\:r=\pm3$
$i.e.,$ The common ratio $=\pm 3$
answered Mar 30, 2014 by rvidyagovindarajan_1
 

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