00:01
Let's say we wanted to derive a formula for the sum of the first n -odd numbers.
00:08
So we can check to see what the first couple will equal.
00:14
So the first one is just one.
00:16
One plus three is four.
00:19
One plus three plus five will get you to nine.
00:23
One plus three plus five plus seven will get you to 16.
00:27
This one will be 25 and this one will be 36.
00:31
And so you might notice a pattern here where the sum of the first n odd numbers appears to be n squared.
00:42
And you can check this with, say, one, three, five, seven, nine, eleven, and thirteen, and thirteen, and you will see that you get 49.
00:59
And so is there a reason for this? is there a way we can prove it? well, the sum of odd numbers is actually just an arithmetic sequence where the common difference is always two.
01:13
So we know the formula for the sum of an arithmetic sequence is n over two times a1 plus a .n.
01:25
So in this case, we keep the n over two, a1 is just one, and an is the nth odd number...