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[music] Since I know about triangular
numbers and I know how to sum constants, I

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can combine those two facts to some more
complicated expressions.

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For example, let's do the sum as n goes
from 1 to k of 2n minus 1.

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All right?
And what this notation mean is that I plug

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in these numbers from 1 to k into this
expression and add them up.

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So, I plug in n equals 1, I get 1.
I plug in n equals 2, I get 3.

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I plug in n equals 3, I get 5.
I plug in n equals 4, I get 7.

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All right.
And I'm going to keep going until I end

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with 2k minus 1.
In words, right, what is this asking me to

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do?
Well, 1, 3, 5, this is adding up odd

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numbers, right?
So, I could say in words that this is the

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sum of the first k odd numbers.
Right?

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I'm going to add up the first k odd
numbers, 1 plus 3 plus 5, until I get to

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the k 5 number, which is 2k minus 1.
Let's first work this out algebraically.

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So, algebraically, I've got the sum of 2n
minus 1 as n goes from 1 to K.

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And this is summing this difference, so I
could write this as the difference of two

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sums.
So, both of these are sums as n goes from

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1 to k, but now I'm adding up 2n, I'm just
adding up 1.

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Now, I can pull out this factor of 2 by
distributivity.

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So, this is 2 times the sum of n as n goes
from 1 to k, minus just the sum of 1.

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N goes from 1 to k.
I have a bit of a formula for the sum of

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just the first k whole numbers, right.
And if you remember back to that formula

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that's k plus 1 times k over 2.
And what's just the sum of 1 as n goes

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from 1 to k?
Well, this is 1 plus 1 plus 1 plus 1 k

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times.
That's just minus k.

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Now, this is 2 times something divided by
2, so this is k plus 1 times k and then,

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minus k.
Well, if I expand this out, I've got k

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plus 1 times k minus 1 times k.
This is k squared.

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So, the sum of the first k odd numbers is
k squared.

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But I can also see this fact that the sum
of odd numbers gives me perfect squares.

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I can see this fact geometrically.
So, I'll draw, say one dot, and then I'll

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draw three more dots.
All right.

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This is 1 plus 3, it's 2 squared.
Then, I'll draw five more dots.

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1 plus 3 plus 5, that's 3 squared.
Then, I'll draw seven more dots.

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All right, and that's 1 plus 3 plus 5 plus
7, that's 4 squared.

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And then, I'll draw nine dots, one, two,
three, four, five, six, seven, eight,

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nine.
And that's five squared.

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1 plus 3 plus 5 plus 7 plus 9 is 5
squared, 25.

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And I'll add 11 dots, one, two, three,
four, five, six, seven, eight, nine, ten,

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11.
And 1 plus 3 plus 5 plus 7 plus 9 plus 11,

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right?
The sum of the first 1, 2, 3, 4, 5, 6 odd

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numbers is 1, 2, 3, 4, 5, 6 squared, it's
36.

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So, the upshot here is that if you sum odd
numbers, you end up getting perfect

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squares.
And, and that's cool independent of

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everything else right, you can see that
fact geometrically.

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But the real take away here is to remember
you can actually do summation problems,

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right?
If somebody gives you a complicated sum,

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by using facts like the sum of just n, and
the sum of constants, you can combine

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those facts to evaluate more complicated
summation problems.