The numeric function is: $$0\cdot5^0, 1\cdot5^1, 2\cdot5^2, \ldots, r\cdot5^r,\ldots$$
My solution is:
$$\begin{align*} &\frac5{1-5z}=1+5z+(5z)^2+\ldots\\\\ &\frac{d}{dz}\left(\frac1{1-5z}\right)= 0 + 1\cdot5 + 2\cdot 5^2 \cdot z +\ldots& [\text{diffentiating wrt }z] \end{align*}$$
Multiply above by $z$ and we get the generating function, which can be written as:
$$z\cdot\frac{d}{dz}\left(\frac1{1-5z}\right) = \frac{5z}{(1-5z)^2}$$
But, the ans given in ans book is:
$$\frac{z}{5(1-(z/5))^2}\;.$$
Is my solution wrong?
Also, please provide solution for these numeric functions as well:
$1, -2, 3, -4, 5,\ldots$
$1, \frac23,\frac39,\frac4{27},\ldots,\frac{r+1}{3^r},\ldots$
$1,1,2,2,3,3,4,4,\ldots$
$0\cdot1, 1\cdot2, 2\cdot3, 3\cdot4,\ldots$