I don't understand the sigma notation and for loop

Question

I have the book saying that

and says that this is equivalent to saying

Answer 1

Looks like the book just says that for given x and y

for (int i = x; i <= y; ++i) Sum += 1;

is similar to

and thus, replacing x = 1 and y = N,

for (int i = 1; i <= N; ++i) Sum += 1;

is similar to

This is of course what you expect: when counting from 1 to N, you count a total of N items. More generally, when counting from x to y, you count y - x + 1 items (+ 1 since you include both boundaries).