Exercise 1.

Find the generating function for each of the following sequences by relating them back to a sequence with known generating function.

1. \(4,4,4,4,4,\ldots\text{.}\)

2. \(2, 4, 6, 8, 10, \ldots\text{.}\)

3. \(0,0,0,2,4,6,8,10,\ldots\text{.}\)

4. \(1, 5, 25, 125, \ldots\text{.}\)

5. \(1, -3, 9, -27, 81, \ldots\text{.}\)

6. \(1, 0, 5, 0, 25, 0, 125, 0, \ldots\text{.}\)

7. \(0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, \ldots\text{.}\)