$P(1)+P(2)+...+P(n)=n^{7}+\lambda$ , $P(\lambda)=?$

Question

$\lambda \in\mathbb{R}$ and $P\in\mathbb{R}[X]$ a polynomial equation

$$P(1)+P(2)+...+P(n)=n^{7}+\lambda,n\in\mathbb{N}^{*}$$

I need to find $P(\lambda)$.

I got $P(n)=n^{7}-(n-1)^{7}$ so $P(\lambda)=\lambda^{7}-(\lambda-1)^{7}$ but the right answer is $1.$How to continue ?

Answer 1

You've calculated $P(x)$ right, so $$\sum_{i=1}^n {P(i)}= \sum_{i=1}^{7}{i^7-(i-1)^{7}}=n^7$$ Therefore $\lambda =0, P(\lambda)=1$