Find the solution to the following lhcc recurrence:
an=3nan-1 for n >or equal to with initial conditions a0=4
Looks like An = 4 * 3^n * n!
To find the solution to the given linear homogeneous constant coefficient (LHCC) recurrence relation, we can follow these steps:
Step 1: Find the characteristic polynomial.
The characteristic polynomial of the recurrence relation is derived by replacing "an" with "r^n", where "r" is a constant, and solving for "r" in the resulting equation. In this case, the recurrence relation is:
an = 3an-1
Replace "an" with "r^n":
r^n = 3r^(n-1)
Divide both sides by r^(n-1) (assuming r is not equal to 0):
r = 3
So, the characteristic polynomial is:
P(r) = r - 3
Step 2: Find the roots of the characteristic polynomial.
To find the roots of the characteristic polynomial, we set P(r) = 0 and solve for "r":
r - 3 = 0
r = 3
There is only one root, which is 3.
Step 3: Write the general solution.
The general solution to the recurrence relation is in the form of:
an = C * r^n
Since we have one root, r = 3, the general solution is:
an = C * 3^n
Step 4: Determine the constant value.
To determine the constant value C, we can use the initial condition given in the problem statement.
Given initial condition: a0 = 4
Replace n = 0 and solve for C:
a0 = C * 3^0
4 = C
Therefore, C = 4.
Step 5: Write the final solution.
Now that we have the constant value C, we can write the final solution to the recurrence relation:
an = 4 * 3^n
So, the solution to the given LHCC recurrence relation is:
an = 4 * 3^n, where n ≥ 0.