For r ≥ 0, the generating function A(z) corresponding to numeric function ar = 4^r+3, is

Concept:

Let (a₀, a₁, a₂, ...., a_r,...) be a numeric function a. Then the infinite series in terms of a parameter z,

a₀ + a₁z + a₂z² +... + a_rz^r + ...  is called generating function of the numeric function a.

a_r is the r^th term of the function.

General term of numeric function is in the from a_r = α^r and the generating function is in the series

1 + αz + α²z² + α³z³+ ..... = \(\frac{1}{1 - \alpha z}\)

A(z) = \(\frac{1}{1 - \alpha z}\)    

Calculation:

We have, a_r = 4^{r + 3} where r ≥ 0

The generating function of a_r is given by 

A(z) = generating funtion of 4^r times 4³

A(z) = 64 × \(\frac{1}{1 - 4z}\)

⇒ A(z) = \(\frac{64}{1- 4z}\)

∴ The generating function of A(z) is \(\frac{64}{1 - 4z}\)