The value of the n+1th term is
In the previous example, the 10th term is 5+3x9=35.
The sum of n+1 first terms ("arithmetic series") can be calculated by grouping the terms two by two :
U0+ U1 + U2 +....+Un-2+Un-1+Un
+Un+Un-1+Un-2+....+ U2 + U1 +U0
=2S=(n+1)x(U0+Un), as Uk+Un-k=U0+ka+U0+(n-k)a=U0+U0+na=U0+Un for all k, hence :
Application : sum of integers from 1 to 1000
It is an arithmetic sequence with a difference of 1, the sum is 1000x1001/2=500500.