Question

desenvolva a 6 linha do triangulo de pascal faça o binomio de newton
(x+k)^6

Answer 1

$\frac{1.x^{6}k^{0}}{0} + \frac{x^{5}k^{1}}{1} + \frac{x^{4}k^{2}}{2} + \frac{x^{3}k^{3}}{3} + \frac{x^{2}k^{4}}{4} + \frac{x^{1}k^{5}}{5} + \frac{x^{0}k^{6}}{6}$
Multiplicando o 1 na frente do x vezes o expoente do x fica dividindo pelo denominador do proximo;
$x^6} + 6x^5.k^1 + 15x^4.k^2 + 20^3.k^3 + 15x^2.k^4 + 6x^1.k^5 + k^6$

Answer 2

A $n-$ésima linha do triângulo de pascal é:

$\dbinom{n}{0}~\dbinom{n}{1}~\dots~\dbinom{n}{n-1}~\dbinom{n}{n}$

A sexta linha é:

$\dbinom{6}{0}~\dbinom{6}{1}~\dbinom{6}{2}~\dbinom{6}{3}~\dbinom{6}{4}~\dbinom{6}{5}~\dbinom{6}{6}$

Em geral,

$(a+b)^{n}=\dbinom{n}{0}a^{n}b^{0}+\dbinom{n}{1}a^{n-1}b^1+\dots+\dbinom{n}{n-1}a^1b^{n-1}+\dbinom{n}{n}a^{0}b^{n}$.
Com isso,

$(x+k)^6=$

$\dbinom{6}{0}x^6k^0+\dbinom{6}{1}x^5k^1+\dbinom{6}{2}x^4k^2+\dbinom{6}{3}x^3k^3+\dbinom{6}{4}x^2k^4+\dbinom{6}{5}x^1k^5+\dbinom{6}{6}x^0k^6$

$(x+k)^6=x^6+6x^5k+15x^4k^2+20x^3k^3+15x^2k^4+6xk^5+k^6$