Question

o coeficiente de x5 no desenvolvimento de (x+2)^9 é: a)64 b)126 c)524 d)1.024 e)2.016

Answer 1

Binômio de Newton

$\boxed{\boxed{(a+b)^{n}=\displaystyle\sum\limits_{k=0}^{n}C_{n,k}\cdot a^{n-k}\cdot b^{k}}}$

Termo geral do Binômio de Newton

$\boxed{\boxed{T_{k+1}=C_{n,k}\cdot a^{n-k}\cdot b^{k}}}$

P.S:

$\boxed{\boxed{C_{n,k}=\dfrac{n!}{k!\cdot(n-k)!}}}$
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Escrevendo (x + 2)^9 na forma:

$(x+2)^{9}=\displaystyle\sum\limits_{k=0}^{9} C_{9,k}\cdot x^{9-k}\cdot2^{k}$

Veja que chegaremos em x^5 quando 9 - k = 5:

$x^{9-k}=x^{5}\\\\9-k=5\\\\k=9-5\\\\\boxed{\boxed{k=4~~(quinto~termo)}}$

Usando o termo geral do Binômio de Newton:

$T_{k+1}=C_{9,k}\cdot x^{9-k}\cdot2^{k}~~~\therefore~~~T_{4+1}=C_{9,4}\cdot x^{5}\cdot2^{4}\\\\\\T_{5}=\dfrac{9!}{4!(9-4)!}\cdot16\cdot x^{5}\\\\\\T_{5}=\dfrac{9\cdot8\cdot7\cdot6\cdot5!}{4\cdot3\cdot2\cdot1\cdot5!}\cdot16x^{5}\\\\\\T_{5}=3\cdot7\cdot6\cdot16x^{5}\\\\\boxed{\boxed{T_{5}=2016x^{5}}}$

Portanto, o coeficiente de x^5 é 2.016