please prove this:

Question

If n ∈ N, prove that the coefficient of xⁿ in (1 + x + x²)ⁿ is 1 + n(n - 1)/ (1!)² + n(n - 1) (n - 2) (n - 3)/(2!)² + ......

Answer 1

(1 + x + x²)ⁿ = (1 + x)ⁿ + ⁿC₁(1 + x)^n-1 x² + ⁿC₂(1 + x)^n-2(x²)²+...ⁿC_r(1 + x)^n-r(x²)^r+....ⁿC_n(x²)ⁿ

Coefficient of xⁿ in(1 + x + x²)ⁿ

= Coefficient of xⁿ in (1 + x)ⁿ + ⁿC₁Coefficient of x^n-2 in (1 + x)^n-1 + ⁿC₂ Coefficient of x^n-4 in (1 + x)^n-2+.....

 = 1 + n x ^n-1C_n-2 + \(\frac{n\times(n-1)}2\) x ^n-2C_n-4 +....

= 1 + n(n - 1) + \(\frac{n(n-1)}2\) x \(\frac{(n-2)(n-3)}2\)+.......

= 1 + \(\frac{n(n-1)}{^2}\) + \(\frac{n(n-1)(n-2)(n-3)}{2^2}\)+.....

Hence proved.