Question

find the middle term i n. the expansion of (x+1/x)⁴​

Answer 1

Answer:

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Answer 2

$Given \: Big( x + \frac{1}{x}\Big)^{4}$

$Number \: of \: terms \: in \: the \: expansion \\= n + 1 \\= 4 + 1 \\= 5\: ( Odd )$

$\blue{Middle \: term } = \Big( \frac{ 5 + 1 }{2} \Big)^{th} \: term \\= \Big( \frac{ 6 }{2}\Big)^{th} \: term \\= 3^{rd} \: term$

/* We know that */

$In \: a\: binomial \: expansion \: ( x + y )^{n}$

$\boxed{\pink{ t_{r+1} = ^{n}C_{r} \times x^{n-r} \times y^{r} }}$

$Here, n = 4 , \: r = 2$

$\red{ Middle \: term } \\= t_{3} \\= t_{2+1} \\= ^{4}C_{2} \times x^{4-2} \times \Big( \frac{1}{x}\Big)^{2} \\= \frac{4!}{(4-2)! 2! }\times x^{2} \times \frac{1}{x^{2}} \\= \frac{4 \times 3 \times 2 \times 1}{ 2 \times 1 \times 2 \times 1 }\\= 6$

Therefore.,

$\red{ Middle \: term } \green { = 6 }$

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