1. ## sigma notation

how to write in correct way:

to calculate the sum of odd numbers between 1 and n;

to calculate the sum of even numbers between 2 and n;

n
E (i+2) --> do you agree?
i=2

2. I would say, for even numbers:

$\displaystyle \sum_{i=0}^{n}(2i+2)$

and for odd:

$\displaystyle \sum_{i=0}^{n}(2i+1)$

how to write in correct way:

to calculate the sum of odd numbers between 1 and n;

to calculate the sum of even numbers between 2 and n;

n
E (i+2) --> do you agree?
i=2
$\displaystyle \displaystyle\sum_{i=2}^n(i+2)=4+5+6+.....+(n-1)+n+(n+1)+(n+2)$

$\displaystyle \displaystyle\sum_{i=1}^{\frac{n+1}{2}}(2i-1)=1+3+5+...+n$

or

$\displaystyle \displaystyle\sum_{i=0}^{\frac{n-1}{2}}(2i+1)=1+3+5+...+n$

for n even, sum of the odd numbers is...

$\displaystyle \displaystyle\sum_{i=1}^{\frac{n}{2}}(2i-1)=1+3+5+...+(n-1)$

or

$\displaystyle \displaystyle\sum_{i=0}^{\frac{n-2}{2}}(2i+1)=1+3+5+...+(n-1)$

That's if "between" is meant to include i=1 and i=n, rather than exclude them