Arithmetic sequences have the very nice property that the average of all numbers in the sequence is the same as the average of the first and last numbers!
For example, the average value in the sequence 4, 7, 10, 13, 16 (an arithmetic sequence with first number 4 and "common difference 3) is (4+ 7+ 10+ 13+ 16)/5= 50/5= 10 (the middle number in the sequence of course) and (4+ 16)/2= 20/2= 10.
Notice that Grandad's formula, is, exactly, n times , the average of the first and last terms.
... and it's very easy to see why: just write the sequence down twice, one underneath the other - but the second one in reverse order. Then add each term to the one underneath it. You get the same total times over. Multiplying this total by , and dividing the result by , will then give the sum of the sequence, of course.
Grandad