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3. The $5 ^{th} , 8 ^{th} \: \:and \: \: 11 ^{th}$ terms of a G.P. are p, q and s, respectively. Show that $q ^2 = ps$

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To prove : $q ^2 = ps$

Let first term=a and common ratio = r

$a_5=a.r^4=p..................(1)$

$a_8=a.r^7=q..................(2)$

$a_1_1=a.r^1^0=s..................(3)$

Dividing equation 2 by 1, we have

$\frac{a.r^7}{a.r^4}=\frac{q}{p}$

$\Rightarrow r^3=\frac{q}{p}$

Dividing equation 3 by 2, we have

$\frac{a.r^1^0}{a.r^7}=\frac{s}{q}$

$\Rightarrow r^3=\frac{s}{q}$

Equating values of $r^3$ , we have

$\frac{q}{p}=\frac{s}{q}$

$\Rightarrow q^2=ps$

Hence proved

Posted by

seema garhwal

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