Sigma notation for a sum of squares of odd numbers

The odd numbers $1, 3, 5, \dots, 15$ can be written as $2i - 1$ for $i = 1, 2, \dots, 8$, since $2(1)-1=1$ and $2(8)-1=15$. So the sum of their squares is $\sum_{i=1}^{8} (2i-1)^2.$

Evaluating term by term: $1^2+3^2+5^2+7^2+9^2+11^2+13^2+15^2 = 1+9+25+49+81+121+169+225$. Adding these values gives $680$.

Therefore $\sum_{i=1}^{8}(2i-1)^2 = 680$.