I need help seeing the work done for this problem in my workbook, so that I can know how to do these on my own in the future. Thanks in advance. You all just take the money and deny questions to be answered. Please allow the tutor to begin the problem or even work an example with different variables so that I can know how to begin and end the problem!!!!!!!!!!!!!!!!!!!!!!!!!!

Consider the following recurrence relation.

*B*(*n*) = 2 if n=1

B(n)= 3**B*(*n* − 1) + 2 if *n* > 1

Use induction to prove that *B*(*n*) = 3^{n} − 1.

(Induction on *n*.) Let *f*(*n*) = 3^{n} − 1.

*Base Case:* If *n* = 1,

the recurrence relation says that *B*(1) = 2,

and the formula says that *f*(1) = 3 − 1 = ,

so they match.

*Inductive Hypothesis:* Suppose as inductive hypothesis that *B*(*k* − 1) =

for some *k* > 1.

*Inductive Step:* Using the recurrence relation,

*B*(*k*) = 3 · *B*(*k* − 1) + 2, by the second part of the recurrence relation

= 3 + 2, by inductive hypothesis

= (3^{k} − 3) + 2

=

so, by induction, *B*(*n*) = *f*(*n*)

for all *n* ≥ 1.

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