Insert two numbers between 3 and 81 so that t | vedclass

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Question

(A) Let $G_{1}$ and $G_{2}$ be two numbers between $3$ and $81$ such that the sequence $3, G_{1}, G_{2}, 81$ forms a $G.P.$Let $a$ be the first term a

Vedclass · Accepted Answer

$9, 27$

Vedclass · Answer

(A) Let $G_{1}$ and $G_{2}$ be two numbers between $3$ and $81$ such that the sequence $3, G_{1}, G_{2}, 81$ forms a $G.P.$Let $a$ be the first term and $r$ be the common ratio of the $G.P.$Here,the first term $a = 3$ and the fourth term $a_{4} = 81$.The formula for the $n^{th}$ term of a $G.P.$ is $a_{n} = a r^{n-1}$.Thus,$81 = 3 	imes r^{4-1} = 3 r^{3}$.$r^{3} = \frac{81}{3} = 27$.$r = \sqrt[3]{27} = 3$.Now,$G_{1} = a r = 3 	imes 3 = 9$.$G_{2} = a r^{2} = 3 	imes (3)^{2} = 3 	imes 9 = 27$.Therefore,the two numbers are $9$ and $27$.

Insert two numbers between $3$ and $81$ so that the resulting sequence is a $G.P.$

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