Finding the nth term of a Quadratic Sequence GCSE Higher Maths

in this video we're going to learn how to find the nth term of quadratic sequence to be able to do this well you need to be able to find the term of linear sequence if you're unsure how to do this or want to revise it quickly first of all check the link in my description so for this question we need to find the nth term we're going to begin by finding the differences between each of the terms so to get from 6 to 19 we add 13. to get from 19 to 38 we add 19 from 38 to 63 is add 25 and 63 to 94 is add 31. we call this the first difference now when the sequence was linear this first difference would be constant that is the same number each time so we were always adding or taking away the same amount but in quadratic sequence the first difference won't be the same instead we need to do the second difference imagine the red numbers here are sequence themselves and find the difference between those so to get from 13 to 19 we had six from 19 to 25 we had six 25 to 31 is add six again this is known as the second difference and notice this time the second difference is constant it stays the same we say that if the second difference is constant the sequence is known as quadratic sequence and has quadratic nth term so how do we go about finding this nth term well the first thing we need to note is that the coefficient of squared will be half of the second difference so if we take that second difference which was always 6 and then half it we'll get 3 which is now going to be the coefficient of squared in our answer so the answer will begin three squared sadly we aren't finished yet though if we were to substitute for one we should get the first term so three times one squared but this just gives three our sequence begins with six though so this can't be the whole story what we do next is write another copy of the sequence down like this then underneath this we write the part of the answer that we know so 3n squared we then generate terms for 3n squared to do this first of all write down the square numbers so 1 4 9 16 25 and so on and then just times them all by 3. so 1 times 3 3 four times three twelve nine times three twenty seven sixteen times three forty eight and twenty five times three seventy five what we can see now is our sequence six nineteen thirty eight and so on is where we want to get to but where we're at at the moment is 3n squared which is 3 12 27 and so on so we want to see what we can do to the 3n squared sequence to get back to our sequence to do this we're going to subtract the sequences so 6 take away 3 is 3 19 take 12 is 7 38 take 27 is 11 63 take 48 15 and finally 94 takes 75 19. this is new sequence here but if we added this sequence to 3n squared we know we'd get our sequence it turns out this sequence is linear we can see the difference is 4 so it must be 4n sequence but it doesn't begin with 4 it begins with 3 so we take away 1. what we've shown here is that our sequence can be split into two parts there's quadratic part here given by 3n squared and there's linear part here given by 4n minus one but if we added the blue bits and the red bits together we get to our sequence so all we need to do is take this four minus one and add it to our answer so the nth term is three squared plus four minus one let's try different example then let's start by finding the differences to get from four to seven we had three seven to fourteen is add seven fourteen to twenty five add 11 and 25 to 40 at 15. this is the first difference and it's not constant so it's not linear sequence so we'll check the second differences three to seven is at four seven to eleven is at four eleven to fifteen is at four this is the second difference and it is constant so it must be quadratic sequence we then take the second difference which is four and half it half of four is two so the answer must start with two squared we then write another copy of the sequence out like this and then we're going to write two squared out to do this start with the square numbers and then times them all by two so one times two two four times two eight nine times two eighteen sixteen times two thirty two and twenty five times two fifty we then subtract so the sequence take away two squared four take two is two seven take eight is negative 1 14 take 18 is negative 4 25 take 32 negative 7 and 40 take 50 negative 10. we've now got the linear sequence in red so we find the nth term of this the difference between terms is take away 3 so it must be minus 3n sequence and since we start at 2 we need to add 5 since negative 3 and 5 is 2. we can then bring the two parts of the sequence together the quadratic part was 2n squared but the linear part was minus 3n plus 5. and that's our nth term thank you for watching this video hope you found it useful check out what think you should watch next and also subscribe so you don't miss out on future uploads