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This video is going to be about finding the sum of an arithmetic sequence and
actually I'm going to deal with this topic in two videos.
In this first one I want to do a basic approach and then in the second one
I'll deal with some examples that you might have some difficulty setting up. So here's what
the problem is about.
Let's say I've got this sequence of numbers that I'm adding together. I've got
1 + 2 + 3 + 4 + 5 + 6.
Now of course I could add these numbers together in the traditional manner,
but let's if we can find a general way to do this so that if we had
a lot of numbers, like say 1 to 100,
we could add all of those together also.
So here's the way we're gonna do it.
If I take the first number, which is 1, in the last number, 6,
and I add them together,
well, the sum of 1 and 6 is 7.
And if I take the second number, 2,
and add that to the second number from the end, which is 5,
I also get a 7.
and if I take the third number,
3,
and I add that
to the third number from the end,
which is 4,
I also get 7.
So,
I've got three 7's here, and 3 times 7 times seven
would be 21. So it seems like my answer should be 21.
Let's add these numbers together in the traditional manner and make sure that's
right.
So 1 plus 2 is 3, 3 plus 3 is 6, three six
6 plus 4 is 10,
10 plus 5 is 15,
and 15 plus 6 is 21.
So this seems to work.
Now let's see if we can find a general rule
for what we did.
The first thing was I took the first number and the last number and added them
together.
So to make this a general rule, instead of saying "the last number", let's call that
number 'n'.
So in other words, I added together
n, the last number,
and the first number.
so that's n plus 1.
And then,
i had to figure out how many pairs of
those numbers I had, in other words, the 1 + 6, 2 + 5, and 3 + 4.
And, basically, I've got half as many pairs
as I have numbers.
So,
if I have n numbers,
then the number of pairs I have, is n over 2.
And if I multiply that n over 2 times the n + 1, I should get
the result I want.
So let's plug in our real numbers into this formula,
n times n+1 over 2.
So let's see... n is 6,
n+1, that's 6 + 1,
and I'm dividing that by 2.
6 divided by 2 is 3.
6 + 1
is 7.
And 3 times 7
is 21.
Now let's try a variation just to be sure this is going to keep working.
So in this first example I had
pairs of numbers. What if i had an odd number of numbers? What if I have
1 + 2 + 3 + 4 + 5? Well, I can't make make
pairs out of everything. I can make a pair out of 1 and 5,
I can make a pair of 2 and 4,
but the 3 is gonna be left by itself.
So let's take the general formula and see if it works when we have an odd number.
So the formula is going to be n times n+1
divided by 2.
In this case, n is 5.
So I'll have 5
times
5 + 1
divided by 2.
5 times 5+1... Well 5 plus 1 is 6,
and 5 times 6 is 30. So I have 30 divided by 2. divided by two
And that equals 15.
And we can add these numbers up make sure it equals 15. 1 + 2
is 3,
plus 3 more is 6,
6 plus 4 is 10,
and 10 plus 5 is 15.
So,
I think that we can see that this formula is going to work for both an odd
and an even number of terms.
Let's try the formula on a big number, something you would want to do by adding in the traditional
way.
So here I've got the sequence 1 + 2 + 3 + 4
and then it goes up to 200.
So
my formula says
n times n+1
divided by 2 will give me the sum.
'n' is 200, that's the last number I have, perhaps
and n+1 will be 200 + 1,
and n+1 will be 200 + 1,
and I'm going to divide that by 2.
Okay, so
200 divided by 2 is going to be 100,
and then I'll multiply that times 200 + 1, which is 201.
When I multiply by 100, all I want to do is
add two zeros to the other number.
So that means I'll have
2 0 1 0 0
2 0 1 0 0,
or 20,100
as the sum that I would get
if I added up the numbers from 1 to 200.
So let's just review this method quickly and make sure you understand the logic behind it.
What I'm doing is I'm thinking of these numbers as pairs of numbers, and each
pair is the same as if I add the first number and the last number,
or the 'n' number, the last one,
plus 1. So that's the n+1.
I've got half as many pairs as I have numbers,
so I'll take the n and divided it by 2 to find out how he pairs I have.
That will give me
n over 2 times n+1,
n times n+1 over 2. I can read it either way.
I plug in the numbers and I get the sum.
Okay. So this works nicely when my first number is 1
and I just increase by 1 each time.
We're gonna run into some difficulties if the first number is not 1.
What if I started with 5 and wanted to go to 200?
And we'll also run into difficulties if instead of increasing by 1 each time,
I increase by, let's say, 2 or 3 or 4. So if i wanted to add
6 + 8 + 10 + 12 and so on,
I might have to
do a little work on this general formula to get it to work for situations like that.
So I'll do that in the second video.
Stick around,
I'll be back soon.
Take care.