Chapter 11.5, Problem 14ES

To determine

(a)

Use iteration to guess an explicit formula for $b_n.$

To determine

(b)

Prove that if k is an integer and x is a real number with $3^k\le x<3^{k+1},\text{ then }\lfloor {\log }_{3}x\rfloor =k.$

To determine

(c)

Prove that for all integers $m\ge 1,$

$\lfloor {\log }_3\left(3m\right)\rfloor =\lfloor {\log }_3\left(3m+1\right)\rfloor =\lfloor {\log }_3\left(3m+2\right)\rfloor .$

To determine

(d)

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