Number sequence 1 2 5 10 what 2 numbers are next and why?

Answer 1

We can work out the next two numbers by getting the nth term. To do this find the difference detween each of the terms (1,2,5,etc...): (differences in subscript) 1 1 2 3 5 5 10 We see that the differences at the first level are not the same; so we do it again, but for the differences, not the terms: 1 2 3 2 5 Now the differences are the same. This tells us that the nth term contains n2. This also tells us that the equation will be in the form An2+Bn+C. If we take n2 as the nth term for now and fill out the first four places we get: 1,4,9,16 which isn't correct. Now we have to find out how to get from n2 to the correct sequence. To do this we look at the difference between the two, these are: 0,2,4,6. We can fill in the rest of the equation (Bn+C) here with ease. If we turn the above differences into a sequence and find its nth term we will have the rest of the equation for the original sequence. To do this we look at the differences as in the original. These are: 0 2 2 2 4 2 6 This is very helpful because the differences are the same. We now know that 2n is involved. However 2n isn't the correct nth term as 1x2 is 2, not 0. To get the correct nth term we substitute the term numbers into 2n and find the difference between the two. The difference is -2 in every case. So know we know that the nth term for the difference between n2 and the original sequence is -(2n-2). That gives us the nth term of n2-(2n-2), however this is messy so we want to expand the bracket. When we expand it we get n2-2n+2 (the -2 changes to +2 because -(2n-2) is the same as -1(2n-2), therefore when expanded we have -1x-2 = -2). So now we know that the nth term is n2-2n+2 we can easily find the next two numbers by substituting in the values 5 and 6. Doing this gives us 17 and 26. The sequence with the next two values is: 1, 2, 5, 10, 17, 26 N.B. Although the next two values were obvious at the very 1st stage, I decided to show you the longer method to help you in future when the solution is not as obvious. -Iban