"Max" wrote...
One way:

Assuming series / numbers is in row1

Sum of Even:
=SUMPRODUCT(--(MOD((1:1),2)=0),1:1)

Sum of Odd:
=SUMPRODUCT(--(MOD((1:1),2)=1),1:1)
....

The original question looked like it could have been solved using standard
variations on Gauss's formula.

Sum of odd numbers from 1 to N = ((N + 1)/2)^2 for N odd.

Sum of even numbers from 2 to N = N * (N + 2)/4 for N even.