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General Form 40

all right in this video I'm going to show you how to create a formula from a given sequence so suppose we've got this sequence so we begin with a seven then it goes to ten then it goes to 13 then it goes to 16 and then we've got a couple of blanks and so it's growing in a very organized predictable way and suppose we wanted to know what the 100th term would be so seven is the first term ten is the second term third term fourth term so we can see that the fifth term would be sixteen plus three which is nineteen and we can see that the sixth term is going to be nineteen plus three is twenty-two and we could keep going like this and we can go up all the way to the 100th term and that's how we can get it but boy it should be nice if we could come up with a formula that would give us the answer in a much quicker way so that's what this video is about so what you're going to do is we're going to take this sequence and let's put it into a nice table so we're going to call this term and we're going to call it one two three four five and by then we're going to see the pattern and we're going to be able to write the formula and then they'd be like the number we'll put here the number so what I mean by this is the first term is seven the second term is ten third term is thirteen and sixteen then nineteen okay and then we want to go up to let's say the nth term is our formula that's the formula we want so the idea is in an arithmetic sequence what's called arithmetic sequence because we keep adding by three each time we're adding by three add by three and add by three and if you think about it what's a faster way of adding by three repeatedly so we're going to go up to the hundredth term and so we're going to have to keep adding a three add three add three add three so what's a faster way of adding a bunch of threes multiplying by three so the formula is begin with n times three this n times three is because we keep repeatedly adding threes so faster way of adding is by multiplying by three okay so now here's the formula the start of it n times three so we look over here and we say well let's try one so we got one times three is three but we didn't want three we wanted seven let's try it on - so we're going to do 2 times 3 because the formula is right here 2 times 3 and 2 times 3 is 6 but we didn't want 6 we wanted 10 and by now we should see the pattern that if you get 3 but we wanted 7 if we get 6 but we wanted 10 that means what this formula needs is a little bit of an adjustment we need to add 4 to the formula let's test it out let's try it on 5 so we're going to do 5 times 3 plus 4 so 5 times 3 is 15 15 plus 4 is 19 and that is exactly the answer we were hoping for so that tells us that this formula indeed is the correct formula which means we can evaluate that formula for a hundred we can finally get our answer for the 100th term so we can insert 100 into our little formula here so we get 100 times 3 plus 4 and so of course a hundred times three is three hundred add 4 gives us three hundred and four so that means what is the value for the 100th term we can officially say it is three hundred and four so in our little chart here we could put three hundred and four so now on this formula sequence arithmetic sequence we have a three then it goes to ten then it goes to 17 then we've got a couple of blanks and we'd like to know well what would be the value of the 100th term so first we have to make sure we understand the pattern how is this sequence growing well 3 to 10 is by adding by 7 7 to 7 to 10 to 17 is adding by 7 which means we're going to always be adding by 7 so we get 24 and we get 31 and we can keep going so if we want the 100th term we could keep adding by 7 until we get all the way up to the 100th term but instead let's make a formula so we're going to make a sequence chart we're going to call this term call this number 1 2 3 4 5 1 2 3 4 5 the first term is 3 second term is 10 third term is 17 then 24 and 31 and then from that that's going to give us the information for the nth term so let's take a look how is it growing keeps adding by 7 every single time add by 7 add by 7 add by 7 so that automatically tells us that our formula is going to be n times 7 because what's a faster way of adding up all of these 7s multiplying by 7 then we're going to try out this formula and see if it needs a little a little adjusting so let's start with 1 so we've got 1 times 7 is 7 but we didn't want 7 we wanted 3 let's try 2 so 2 times 7 equals 14 but we didn't want 14 we were hoping for a 10 so it seems like every single time our formula gives us an answer that's too big so we need to adjust so we're going to adjust by subtracting by 4 let's test it out so let's try it out on the fifth one so we're going to do 5 times 7 minus 4 so 5 times 7 is 35 minus 4 31 sure enough that's exactly what we were hoping for so that tells us we've got the correct formula so now we can use this formula to figure out what the 100th term is so we put a hundred up here and then we evaluate this formula for a hundred so we do a hundred times seven minus four so 100 times seven is seven hundred minus 4 gives us six hundred ninety-six so we know the 100th term in this sequence is six hundred and ninety-six