00:01
Okay, so here we go, we've got to find these sums.
00:05
So the first one is find the sum of the positive three digit numbers whose last digit is 3.
00:12
Alright, so that would be 103 plus 113 plus 123 plus 133 and then it keeps going and the last one is going to be 993, right? the one before that would be 983.
00:36
Okay, and i want to focus in on these guys right here, right? so if i take, it's really if i just kind of forget about the threes, right? i'll add all those threes later but this is like i have a 10 and an 11 and a 12 and a 13 and it keeps going and then a 98 and a 99.
01:04
That's supposed to be 98.
01:05
Okay, so and we're adding all of these up.
01:09
Now i know what you're thinking, well that's really 100 not 10.
01:14
Okay, fine, i'll just take this whole thing and multiply it by 10.
01:18
If i factor the 10 out then i have 10 plus 11 plus 12.
01:21
So when i add these up, all of these from 10 to 99, like if i add that together i get 109 and if i add 11 and 98 i get, oh look, 109 and if i add 12 and it'd be 88, not 88, 97 i would get 109.
01:46
So i get these 109s.
01:50
Now how many pairs are there? well from 10 to 99, 10 to 11, there's actually, you might think, oh there's 89.
01:58
Well actually there's 90 of them.
02:01
From 10 to 99 there's 90 numbers there but we paired them all up so we just have to multiply this by 90 divided by 2 or 45 and so now remember though all of this is being multiplied by that 10 that's in front.
02:19
So all of that added together, when you go 10 times 109 times 45, all of that comes out to be 49050.
02:35
Well that's all fine and good but remember these 3s that we kind of lopped off, remember all these guys? we've got to add those on too.
02:44
Those are still there, those are still there.
02:46
So we have to add, well how many 3s are there? well if there were 90 of the numbers, there's 90 of those 3s so we have to add 90 times 3 which is 270 so our answer comes out to be, when you add 270 to that, you get 49320.
03:12
There's your answer right there, the sum of all those numbers.
03:15
Now you might have been thinking, oh my gosh i'm going to have to type in 103 plus 113 plus 120 and type all 90 of these numbers and hopefully not type anything wrong and hopefully you pick all the right ones so it's easy to mess it up.
03:31
So there's always a little trick usually with these and that's the little trick, you've got to realize, oh if i just lop off the 3s and divide out a 10, i got 10 plus 11, oh okay i can add that up.
03:43
It's a little trick for that.
03:44
And i have 93s, that's 270, it's a little trick for that, i can multiply that together, get 270, make sure i added the right number, oh my goodness.
04:03
Yeah there it was, i added 270, okay i thought i had the wrong thing.
04:06
So there you go, you get 90 times 3 and so you can break it up into some smaller problems.
04:15
Alright, the second one says find the sum of all positive odd numbers less than 400 that are divisible by 5.
04:34
There we go, that's what i'm trying to do.
04:37
Okay, cleaning up the board a little bit.
04:45
So all of the odd integers less than 400 that are divisible by 5, well that's going to be 35 and 15, right, because you skipped 10, and 25 and 35 and 45 and we're adding all these together, you know, all the way up until 400.
05:13
So we're not going to count 400, so 395 and the 385 is right here and 375 is right here.
05:26
Divisible by 5, it has to end in a 0 or a 5, well the only odd ones are the ones that end in 5.
05:34
So, just like we did before, if you notice, like if i take these two, the first number and the last number, add those together, it's 400.
05:44
If i take these two, second number and second to last number adds to 400.
05:50
Third number and third to last number, 400.
05:53
So i'm going to get 400, well i've got to figure out how many numbers are there, right? it's divisible by 5, so 400 divided by 5 is 80, but that includes all the even numbers.
06:11
So there should be 40, only 40 of these numbers, so we should get, sorry, 40 divided by 2 because we have 40 numbers but we paired them up, so we only have 20 pairs.
06:30
So the answer should come out to be 8 ,000.
06:36
You add all those up, you should get 20 of those 400s.
06:40
Now if you're worried, you can come up with a little function, it's an arithmetic sequence, our common difference is 10 and we've got to subtract off 5.
06:52
So if you want to figure out what n is for the last number, if you put 1 in, you're going to get 5.
06:59
But if you put some number in for n to get 395, that's how many terms you have.
07:05
So you can go 395 equals 10n minus 5, add the 5, and you get 400 equals 10n, divide by 10.
07:18
So n does equal 40, that's where the 40 comes from, but remember we paired them up, so that's why we had to do 40 divided by 2.
07:27
You only have, you have 40 numbers, but when you put them together to add to 400, you have 20 pairs.
07:33
So there you go, that's how you do that.
07:35
You don't have to, you know, you don't have to go 5 plus 15 plus, you know, you're going to have 40 numbers and that's going to take you forever to put in there.
07:45
Alright, so i'm scrolling down again.
07:49
I got to hit the thing just, oh there we go.
07:55
Alright, third one, it says find the sum of the series, and it goes 1 minus 3 plus 5 minus 7 plus 9 minus 11.
08:04
So i'm going to write it out.
08:10
Make a mess here.
08:12
So 1 minus 3 plus 5 minus 7 plus 9 minus 11, and the thing i want to look at here, it's not an arithmetic sequence or series.
08:37
So, so we have, you know, we have, you know, and it goes all the way dot dot dot to 101.
08:49
Ah, trying to draw on the screen and then if my hand hits weird, or 1000, 1001 is the last one.
09:00
Okay, so the thing i want you to realize is a couple things that, a couple things we can look at.
09:07
So, you know, how do you figure out, you know, is it going to be, is this going to be 101 minus 1000 or minus 1003 or is it 1, alright, 999, 999 minus 1001.
09:42
Well i'm going to guess it's this.
09:44
I'm guessing it's that right there because it has a plus sign in front of it right here.
09:50
All the rest of it had a minus sign.
09:52
So i'm going with the assumption that it would be 1001 minus 103.
10:01
Now why would i do all of that? well, let's look at what happens here.
10:07
This right here makes a negative 2.
10:10
This right here, negative 2.
10:15
Alright, and we're adding all of these up.
10:18
In fact, even this right here is a negative 2.
10:21
So now what we have to do is we just have to figure out how many of these negative 2's do we have.
10:28
How many negative 2's do we have? okay, well if you notice it goes like, forget about the pluses and minuses, it goes 1, 3, 5, 7.
10:39
So our function, our function here is, the common difference is 2, so 2n.
10:49
We have to subtract off 1, so when we put 1 in we'll get 1.
10:53
We put 2 in we get 3.
10:55
Okay, so if i set this equal to 1001, 2n minus 1, and i add the 1 and divide by 2, 1002 equals 2n, so 501 equals n...