Question

P=$\frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-c)(b-a)}+\frac{c^n}{(c-a)(c-b)}$
=$\frac{a^n(b-c)+b^n(c-a)+c^n(a-b)}{(a-b)(b-c)(c-a)}$

Q=$a^n(b-c)+b^n(c-a)+c^n(a-b)$

n=1,P=0
n=2,P=1
n=3,P=a+b+c
n=4,P=$a^2+b^2+c^2+ab+bc+ca$
n=n,P=???

Generalize it, please.

Answer 1

Answer:

$\begin{gathered} {\left( {a + 2b - 3c} \right)^3} + {\left( {b + 2c - 3a} \right)^3} + {\left( {c + 2a - 3b} \right)^3} \hfill \\ \quad = 3\left( {a + 2b - 3c} \right)\left( {b + 2c - 3a} \right)\left( {c + 2a - 3b} \right). \hfill \\ \end{gathered}$