Find an of {an} given a1 = 1, and an+1 = Sn + 2n + 1
Hello, MATNTRNG!
I hope the is a typo . . .
Crank out the first few terms: .
Take the difference of consecutive terms,
. . then the differences of the differences, and so on.
. .
Since the second differences are constant,
. . the generating function is of the second degree, a quadratic.
The general quadratic function is: .
Use the first three values of the function and substitute:
. .
. .
. .
Substitute into [5]: .
Substitute into [3]: .
Therefore, the generating function is: .