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STD 11 - 8. sequence and series — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
Question
If $m$ arithmetic means $( A . Ms )$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that $4^{\text {th }}$ $A.M.$ is equal to $2^{\text {nd }}$ $G.M.$, then $m$ is equal to
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Answer

a
$3, A _{1}, A _{2} \ldots \ldots \ldots . A _{ m }, 243$

$d =\frac{243-3}{ m +1}=\frac{240}{ m +1}$

Now $3, G _{1}, G _{2}, G _{3}, 243$

$r=\left(\frac{243}{3}\right)^{\frac{1}{3+1}}=3$

$\therefore \quad A_{4}=G_{2}$

$\Rightarrow \quad a +4 d = ar ^{2}$

$3+4\left(\frac{240}{ m +1}\right)=3(3)^{2}$

$m=39$

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