Summation Formula as an answer:

Question

I am given the following multi-variable function: $$f(x_1,x_2,x_3,...,x_n)=\sum_\limits{k=1}^nkx_k$$

Find $f(1,1,1,...,1)$

Why is the answer $\dfrac{n(n+1)}{2}$?

Is it because $x_n=1$ thus leaving $\sum_\limits{k=1}^nk\therefore\text{the answer is } \frac{n(n+1)}{2}$

Answer 1

Just expand:

$\begin{align*} f(x_1, \dotsc, x_n) &= \sum_{1 \le k \le n} k x_k \\ f(1, \dotsc, 1) &= \sum_{1 \le k \le n} k \\ &= \frac{n (n + 1)}{2} \end{align*}$

The last by the known formula for the sum of an arithmetic series.