Find-the-middle-term-in-the-expansion-of-x-3-x-9-

Question Number 36166 by Rio Mike last updated on 29/May/18

Find the middle term in

the expansion of {(x^{} + \frac{3}{x})}^{9}

Commented by abdo.msup.com last updated on 30/May/18

w e h a v e {(x + \frac{3}{x})}^{9} = \sum_{k = 0}^{9} C_{9}^{k} x^{k} {(\frac{3}{x})}^{9 - k}

= \sum_{k = 0}^{9} C_{9}^{k} 3^{9 - k} x^{k + k - 9}

= \sum_{k = 0}^{9} C_{9}^{k} 3^{9 - k} x^{2 k - 9} t h e m i d d l e

t e r m i s {a x}^{5} s o w e m u s t h a v e

2 k - 9 = 5 \Rightarrow k = 7 a n d t h e c o e f f c i e n t

i s 3^{9 - 7} C_{9}^{7} = 9 . \frac{9!}{7! 2!} = \frac{9.9.8}{2}

= 81 \times 4 = 324 s o t h e c o e f f i c i e n t i s

354 x^{5} .

t h e c o e f f i c i e n t i s 324 x^{5} .