Arithmetic sequence - Arithmetic series

An arithmetic sequence (or progression) is obtained by adding a constant "a" to the previous term :
Un+1=Un+a
"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+1th term is

Un=U0+na
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 :

S=(n+1)(U0+Un)/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.

 

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