Question

$\rm(x + y) {}^{n} = {x}^{n} + nx {}^{n - 1} y + \dots + {y}^{n}$​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\rm{\left(x+y\right)^{n}}$

Using binomial expansion,

$\rm{\implies\,\left(x+y\right)^{n}={\,}^{n}C_{0}\left(x\right)^{n-0}\left(y\right)^{0}+{\,}^{n}C_{1}\left(x\right)^{n-1}\left(y\right)^{1}+{\,}^{n}C_{2}\left(x\right)^{n-2}\left(y\right)^{2}+\cdots+{\,}^{n}C_{n}\left(x\right)^{n-n}\left(y\right)^{n}}$

$\rm{\implies\,\left(x+y\right)^{n}={x}^{n}+n\left(x\right)^{n-1}\left(y\right)+\dfrac{n(n-1)}{2}\left(x\right)^{n-2}\left(y\right)^{2}+\cdots+{y}^{n}}$