How to calculate the sum of numbers on the interval from a to b, without a loop?

Question

Mathematics

Faced this problem for a long time and still does not give rest. It is clear that it is time-consuming to solve this with a cycle, but I also found another way out - an arithmetic progression, but here it starts not from 1, but from an arbitrary element, so here the standard formula will not work.
I specify the question:
How to mathematically calculate the sum, for example, from 47 to 72, i.e. 47 + 48+ 49 ... + 72 ?

Answer 1

Here is the standard formula.
S = (a1+an)*n/2, where a1 is the first member, an is the last, n is the number of members of the sequence
We have (47+72)*26/2 = 1547
Total your formula: S = (a+b) (b-a+1)/2

Answer 2

result = ((b+1)*b-(a-1)*a) div 2;
a=1 b=100 result=5050 (that's how much it should be, not 1050)
a=2 b=5 result=14 (2+3+4+5=14)
a=5 b=10 result=45 ( 5+6+7+8+9+10=45)

Answer 3

How to mathematically calculate the sum, for example, from 47 to 72, i.e. 47 + 48+ 49 ... + 72 ?

Calculate the sum of the first 72 terms of the arithmetic progression and subtract from it the sum of the first 46 terms of the arithmetic progression.