I want you to answer Part B only, the answers for Part A has been provided. I have also provided you with the explicit formula for Part B$.$ I need you to provide the steps in how that formula was found. Please do not copy and paste a previously answered question similar to this.

Given the sequence defined with the recurrence relation:

$b_0=2$

$b_k=3b_{k-1}+2k,$AAkinZ$|k|1$

A$.$ Calculate the value for $b_1,b_2,b_3,b_4.$ Keeping the intermediate steps will help find the pattern and solve for the next part.

B$.$ Using iteration, solve the recurrence relation when $n0($i$.$e$.$ find an analytic formula for $b_n).$ Simplify your answer as much as possible, showing your work. In particular your final answer should not contain ${\displaystyle \underset{?}{\overset{?}{}}}$ and $.$ You should not use any non$-$arithmetic operations like ceil, floor, round, or any if structure. You must show your work to get full marks.

Part A:

$b_0=2$

$b_k=3b_{k-1}+2k,$AAkinZ,$|k|1$

$b_1=8$

$b_2=28$

$b_3=90$

$b_4=278$

Dart B

$b_n=2*3^n+\frac{1}{2}(3^{n+1}-2n-3)$