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  • For a positive integer , if the expansion <img alt="\left(\frac{5}{y^2}+y^4\right)^n" src="https://entrancecorner.on

For a positive integer n, if the expansion \left(\frac{5}{y^2}+y^4\right)^nhas a term independent of y, then n can be taken as

 

Option: 1

21


Option: 2

18


Option: 3

15

 


Option: 4

All the above


Answers (1)

best_answer

Let (r+1)^{t h}term of  \left(\frac{5}{y^2}+y^4\right)^nbe independent of y ,then 

\begin{aligned} & T_{r+1}=n C_r\left(\frac{5}{y^2}+y^4\right)^n \\ & T_{r+1}=n C_r\left(\frac{5}{y^2}\right)^{n-r}\left(y^4\right)^r \\ & T_{r+1}=n C_r 5^{n-4} y^{6 r-2 n} \end{aligned}

The above term to be independent of y

\begin{aligned} & 6 r-2 n=0 \\ & n=3 r \end{aligned}

All the multiples of 3, satisfies the value of n.

Posted by

HARSH KANKARIA

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