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Question

Find $$p$$ if the $$4^{th}$$ term is $$\dfrac 52$$


A
0.5
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B
0.3
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C
0.25
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D
4
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Solution

The correct option is A $$0.5$$
Given that: $$T_4=\cfrac52$$
Solution:
$$4^{th}$$ term i.e. $$T_4$$ in the expansion of $$\left(px+\cfrac1x\right)^n={}^n C_3\times(px)^3\times\left(\cfrac1x\right)^{n-3}$$
Now, $$T_4$$$$={}^n C_3\times(px)^3\times\left(\cfrac1x\right)^{n-3}$$
or, $$T_4$$$$={}^n C_3\times p^3\times x^{6-n}$$
Since, $$T_4$$ is indepemdent of $$x.$$
So, power of $$x$$ i.e. $$6-n$$ should be zero.
or, $$6-n=0$$
or, $$n=6$$
Now, $$T_4={}^6C_3\times p^3=\cfrac52$$
or, $$20\times p^3=\cfrac52$$
or, $$p=\cfrac12=0.5$$
Hence, A is the correct option.

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