Summation involving 2 variables

Question

I am trying to understand how to expand a summation equation:

$$\sum_{j=1}^3 \sum_{i = j + 1}^4 (25-5i)$$

how do I expand the inner equation involving $i = j+1$ ?

Thanks!

Answer 1

$$ \sum_{i = j + 1}^4 (25-5i)=\sum_{i=1}^4(25-5i)-\sum_{i=1}^j(25-5i)$$

Can you take it from there?

Answer 2

Alternatively one can swap the order of summation

$$5\sum_{j=1}^3 \sum_{i=j+1}^4 5 - i = 5 \sum_{i=2}^4 \sum_{j=1}^{i-1} 5 - i = 5 \sum_{i=2}^4 (5-i)(i-1) = 5\left[3\cdot 1 + 2\cdot 2 + 1\cdot 3\right] = 50$$

since the summand is independent of $j$.