Textbook Problem

Exercises 12—15 refer to the following algorithm segment. For each positive integer n, let $b_n$be the number of iterations of the while loop.

$\begin{array}{l}\text{while}\left(n>0\right)\\ n:=ndiv3\end{array}$

end while
14. a. Use iteration to guess an explicit formula for $b_n$ .

b. Prove that if k is an integer and x is a real number with $3^k\le x<3^k$then $\lfloor {\log }_{3}x\rfloor$ .
c. Prove that for every integer $m\ge 1,\lfloor {\log }_3\left(3m\right)\rfloor =\lfloor {\log }_3\left(3m+1\right)\rfloor =\lfloor {\log }_3\left(3m+2\right)\rfloor$ .

d. Prove the correctness of the formula you found in part (a).