U

"a" is named the "difference" of the arithmetic sequence.

For instance starting from 5 with a difference of 3 : 5, 8, 11, 14, 17, 20, 23, 26, 29, 33, 35, ...

The value of the n+1^{th} term is

U_{n}=U_{0}+na

In the previous example, the 10^{th} term is 5+3x9=35.

The sum of n+1 first terms ("arithmetic series") can be calculated by grouping the terms two by two :

U_{0}+ U_{1} + U_{2} +....+U_{n-2}+U_{n-1}+U_{n}

+U_{n}+U_{n-1}+U_{n-2}+....+ U_{2} + U_{1} +U_{0}

=2S=(n+1)x(U_{0}+U_{n}),
as U_{k}+U_{n-k}=U_{0}+ka+U_{0}+(n-k)a=U_{0}+U_{0}+na=U_{0}+U_{n}
for all k, hence :

S=(n+1)(U_{0}+U_{n})/2

Application : sum of integers from 1 to 1000

It is an arithmetic sequence with a difference of 1, the sum is 1000x1001/2=500500.