Find the closed form of the generating function of the sequence an=3n−4 , n=0, 1, 2, ….

Sure, here are the steps to find the closed form of the generating function of the sequence an=3n−4 , n=0, 1, 2, ….

Step 1: Define the sequence The sequence is defined as an=3n−4 for n=0, 1, 2, …

Step 2: Write down the generating function The generating function G(x) of a sequence {an} is defined as G(x) = Σ (from n=0 to ∞) an*x^n.

So for our sequence, the generating function is G(x) = Σ (from n=0 to ∞) (3n-4)*x^n.

Step 3: Break it down We can break this down into two separate series: G(x) = 3Σ (from n=0 to ∞) nx^n - 4*Σ (from n=0 to ∞) x^n.

Step 4: Find the closed form of each series The first series is a known series, the derivative of the geometric series. Its closed form is 1/(1-x)^2.

The second series is a geometric series, its closed form is 1/(1-x).

Step 5: Substitute back Substitute these closed forms back into the equation from step 3 to get the closed form of the generating function: G(x) = 3/(1-x)^2 - 4/(1-x).

So, the closed form of the generating function of the sequence an=3n−4 , n=0, 1, 2, … is G(x) = 3/(1-x)^2 - 4/(1-x).