Linear non-homogeneous recurrence relation problem

Question

Solve the recurrence relation $$a_n = 2a_{n−1} + 3 · 2^{n}, a_{0} = 5 $$.

Can someone help me understand this question with a simple step? I don't need a high level solution, any basic step will be appreciated. Thank you.

Answer 1

Hint:   divide by $\,2^n\,$, then $\dfrac{a_n}{2^n} = \dfrac{a_{n−1}}{2^{n-1}} + 3\,$, which shows that $\,\dfrac{a_n}{2^n}\,$ is an arithmetic progression.

Answer 2

• Solve the associated homogeneous recurrence relation $\; a_n=2a_{n-1}$, which is just a geometric series $a_n=K2^n$ for some constant $K$.
• Find a particular solution of the complete equation $\;a_n=2a_{n−1} + 3 · 2^{n}$. As $2^n$ is a solution of the homogeneous equation, you can seek for a particular solution of the form $$ a_n=A\mkern 1mun\cdot 2^n$$ for some constant $A$.
• Add the particular solution of the complete equation to the general solution of the homogeneous equation, and adjust the constant $K$ so that the initial condition be satisfied.