The rth term of an arithmetic progression is 1+4r. Find, in terms of n, the sum of the first n terms of the progression?
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.