Sum of a multi index series (really dumb question)

Question

$$\sum_{\substack{i,j=1 \\ i \neq j}}^{l} (x_iy_i + x_jy_j) = k \sum_{i=1}^l x_iy_i$$

I have to find $k$. I know the question is really stupid, but for some reason I am unable to solve this.

Answer 1

Hint: add and remove the forbidden ($i\ne j$) terms.

$$ \sum_{\substack{i,j=1 \\ i \neq j}}^{l} x_iy_i= \sum_{\substack{i,j=1}}^{l} x_iy_i-\sum_{\substack{i,j=1 \\ i = j}}^{l} x_iy_i= \sum_{j=1}^l\sum_{\substack{i=1}}^{l} x_iy_i-\sum_{i=1}^{l} x_iy_i=(l-1)\sum_{i=1}^{l} x_iy_i.$$

Same for $x_jy_j$.

Answer 2

Hint: For example, either $i=1$ or $j$ may be equal to $1.$ So, how many terms containing $x_1y_1$ will there be?

If you want to get an idea, start by looking at various small values of $l,$ and computing the left-hand side explicitly.

For the proof, contrast with $$\sum_{I,j=1}^l(x_iy_I+x_jy_j),$$ which is easier to calculate.