Find-the-coefficient-of-x-6-in-2x-1-6-x-2-x-1-4-4-

Question Number 58373 by Tawa1 last updated on 22/Apr/19

Find the coefficient of x^{6} in {(2 x + 1)}^{6} {(x^{2} + x + \frac{1}{4})}^{4}

Commented by maxmathsup by imad last updated on 24/Apr/19

w e h a v e {(2 x + 1)}^{6} {(x^{2} + x + \frac{1}{4})}^{4} = 2^{6} {(x + \frac{1}{2})}^{6} {(x + \frac{1}{2})}^{8}

= 2^{6} {(x + \frac{1}{2})}^{14} = 2^{6} \sum_{k = 0}^{14} C_{14}^{k} x^{k} {(\frac{1}{2})}^{14 - k} = \frac{2^{6}}{2^{14}} \sum_{k = 0}^{14} C_{14}^{k} 2^{k} x^{k} s o t h e c o e f f i c i e n t

o f x^{6} i s a_{6} = 2^{- 8} C_{14}^{6} 2^{6} = \frac{1}{4} C_{14}^{6} = \frac{1}{4} \frac{14!}{6! 8!} .

Answered by tanmay last updated on 22/Apr/19

{(2 x + 1)}^{6} {[{(x + \frac{1}{2})}^{2}]}^{4}

{(2 x + 1)}^{6} {(x + \frac{1}{2})}^{8}

o r m e t h o d

{(2 x + 1)}^{6} {[{(x + \frac{1}{2})}^{2}]}^{4}

= 2^{6} {(x + \frac{1}{2})}^{6} {(x + \frac{1}{2})}^{8}

= 2^{6} {(x + \frac{1}{2})}^{14}

= 2^{6} \times 14 c_{8} \times x^{6} \times {(\frac{1}{2})}^{8}

= \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9}{6 \times 5 \times 4 \times 3 \times 2 \times 2^{2}}

= \frac{14 \times 13 \times 11 \times 9}{6 \times 4}

= \frac{7 \times 13 \times 11 \times 3}{4} = \frac{3003}{4}

Commented by Tawa1 last updated on 23/Apr/19

God bless you sir

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