What is 1^2 + 3^2 + 5^2 + ... + 23^2?

Question

1²+3²+5²+...+23²=?

Answer 1

You can do this by brute force. If you write out the entire summation, you get

1 + 3^2 + 5^2 + 7^2 + 9^2 + 11^2 + 13^2 + 15^2 + 17^2 + 19^2 + 21^2 + 23^2

because the pattern is that the number being squared goes up by 2. To calculate the sum, work from left to right squaring each term:

1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + 289 + 361 + 441 + 529

When you add up all the terms, you get 2300.

Extra credit: But what if there are hundreds or thousands of terms? It would be exhausting to write them all out and sum them up. It turns out that it's possible to find what we call an "explicit formula" for the summation, which is a formula that uses the number of terms to calculate the sum. To do so is not an easy task for an Algebra 1 student! The explicit formula for this summation is

n + 2n(n-1) + 2n(n-1)(2n-1)/3.

There are 12 terms in the summation 1^2 + 3^2 + ... + 23^2 because (23 + 1)/2 = 12. If you plug 12 for n into the explicit formula, notice that you get 2300.