Hi. Please can you help me? I have three Maths tests next week, and we have a revision sheets which says lots of stuff, and the last one I can't do! It says nth term. I can do nth term all right, but I was wondering if there is an equation for a sequence if each time the gap changes, eg;
8, 11, 15, 20, 26
+3 +4 +5 +6
__n +/- ___.
Thanks. K
Hello, Kate!
Here's another approach . . .
Take the difference of consecutive terms.
Then take the differences of the differences, and so on
. . until you get a row of constants.
. .
The row of constants appeared in the second differences.
. . Hence, the generating function is of the second degree, a quadratic.
There are a number of methods for determining the generating function.
. . Here's one of them . . .
We know that the function has the form: .
Using the first three values of the sequence, we have:
. .
Then solve the system of equations . . .
Subtract [1] from [2]: .
Subtract [2] from [3]: .
Subtract [4] from [5]: .
Substitute into [4]: .
Substitute into [1]: .
Therefore: .