[music] Since I know about triangular numbers and I know how to sum constants, I
can combine those two facts to some more complicated expressions.
For example, let's do the sum as n goes from 1 to k of 2n minus 1.
All right? And what this notation mean is that I plug
in these numbers from 1 to k into this expression and add them up.
So, I plug in n equals 1, I get 1. I plug in n equals 2, I get 3.
I plug in n equals 3, I get 5. I plug in n equals 4, I get 7.
All right. And I'm going to keep going until I end
with 2k minus 1. In words, right, what is this asking me to
do? Well, 1, 3, 5, this is adding up odd
numbers, right? So, I could say in words that this is the
sum of the first k odd numbers. Right?
I'm going to add up the first k odd numbers, 1 plus 3 plus 5, until I get to
the k 5 number, which is 2k minus 1. Let's first work this out algebraically.
So, algebraically, I've got the sum of 2n minus 1 as n goes from 1 to K.
And this is summing this difference, so I could write this as the difference of two
sums. So, both of these are sums as n goes from
1 to k, but now I'm adding up 2n, I'm just adding up 1.
Now, I can pull out this factor of 2 by distributivity.
So, this is 2 times the sum of n as n goes from 1 to k, minus just the sum of 1.
N goes from 1 to k. I have a bit of a formula for the sum of
just the first k whole numbers, right. And if you remember back to that formula
that's k plus 1 times k over 2. And what's just the sum of 1 as n goes
from 1 to k? Well, this is 1 plus 1 plus 1 plus 1 k
times. That's just minus k.
Now, this is 2 times something divided by 2, so this is k plus 1 times k and then,
minus k. Well, if I expand this out, I've got k
plus 1 times k minus 1 times k. This is k squared.
So, the sum of the first k odd numbers is k squared.
But I can also see this fact that the sum of odd numbers gives me perfect squares.
I can see this fact geometrically. So, I'll draw, say one dot, and then I'll
draw three more dots. All right.
This is 1 plus 3, it's 2 squared. Then, I'll draw five more dots.
1 plus 3 plus 5, that's 3 squared. Then, I'll draw seven more dots.
All right, and that's 1 plus 3 plus 5 plus 7, that's 4 squared.
And then, I'll draw nine dots, one, two, three, four, five, six, seven, eight,
nine. And that's five squared.
1 plus 3 plus 5 plus 7 plus 9 is 5 squared, 25.
And I'll add 11 dots, one, two, three, four, five, six, seven, eight, nine, ten,
11. And 1 plus 3 plus 5 plus 7 plus 9 plus 11,
right? The sum of the first 1, 2, 3, 4, 5, 6 odd
numbers is 1, 2, 3, 4, 5, 6 squared, it's 36.
So, the upshot here is that if you sum odd numbers, you end up getting perfect
squares. And, and that's cool independent of
everything else right, you can see that fact geometrically.
But the real take away here is to remember you can actually do summation problems,
right? If somebody gives you a complicated sum,
by using facts like the sum of just n, and the sum of constants, you can combine
those facts to evaluate more complicated summation problems.