Question

Find functional equations for the generating functions whose coefficients satisfy the following relations AND solve the recurrence relation using generating functions.

(a) an = an-1 + 2, a0 = 1

Answer 1

The generating function for the recurrence relation
`\(a_n = a_(n-1) + 2\)` with
`\(a_0 = 1\)` is
`\(A(x) = (1)/(1-x)\)`. The solution to the recurrence relation is

`\(a_n = 2n + 1\).`

To find the generating function for the given recurrence relation

`\(a_n = a_(n-1) + 2\)` with
`\(a_0 = 1\)`, let's define the generating function \(A(x)\) as:

`\[ A(x) = \sum_(n=0)^(\infty) a_n x^n \]`

Now, we can express the given recurrence relation in terms of the generating function:

`\[ A(x) = a_0 + a_1x + a_2x^2 + \dots \]`

`\[ A(x) = 1 + (a_0 + 2)x + (a_1 + 2)x^2 + \dots \]`

Now, noticing that
`\(a_n = a_(n-1) + 2\)`, we can express the generating function as:

A(x) = 1 + (A(x) + 2)x

Solving for A(x):

`\[ A(x) = (1)/(1-x) \]`

Now, to solve the recurrence relation, we can expand A(x) into a power series:

`\[ A(x) = 1 + x + x^2 + x^3 + \dots \]`

Thus, the solution to the recurrence relation
`\(a_n = a_(n-1) + 2\)` with

`\(a_0 = 1\)` is
`\(a_n = 2n + 1\).`