Final answer:
To solve the given recurrence relation an = 2an-1 + 3n, both the homogeneous and particular parts must be solved. After finding the general solutions, the initial condition aj = 5 is applied to determine the specific solution.
Step-by-step explanation:
To solve the recurrence relation an = 2an-1 + 3n, we'll approach it step-by-step:
Identify the type of recurrence relation. In this case, it's a first-order linear homogeneous recurrence relation with a non-homogeneous part, 3n.
Solve the homogeneous part: an = 2an-1. This is a geometric sequence where each term is twice the previous term. If a1 is the base case, then an = 2n-1a1.
To solve the non-homogeneous part, 3n, you would look for a particular solution. A suitable guess might be a linear function of n, say bn + c. Plugging this guess into the recurrence relation and solving for b and c will give you the particular solution.
Add the general solution of the homogeneous part to the particular solution of the non-homogeneous part to get the general solution of the recurrence relation.
Lastly, apply the initial condition aj = 5 to find the specific solution. Substitute j with the first term index and solve for a1, then substitute a1 back into the general solution.
When you combine both the solutions from the homogeneous and particular parts and add the initial condition, you get the complete solution for the given recurrence relation.