Iteration

Given the sequence defined recursively as

0=1b0=1

bk = 31101+33bk110bk1+3 for every integer k>0

You will now use iteration to deduce a solution for this sequence:

Give the first 5 terms of the sequence. Show and keep the intermediate expansions because they are more important than the final values for noticing a pattern (and your grade will depend on it).

Based on this pattern, guess an analytical solution for this sequence. Your final formulas should not include any or operators.

Hint: Note that for any non-zero x, /(/)+a/x(b/x)+c can be simplified by multiplying the numerator and denominator by x.

The pedagogical goal of this question is not to find an answer, but to learn how to use iteration to notice patterns in sequences, which is a more difficult skill.

In order to do this, you must work from intermediate values instead of final values. Dodistribute your operations to remove the parentheses in each term of the sequence, and dosimplify fractions if you can, but do not calculate the results of additions, multiplications, and exponentiations, because if you do the pattern may disappear.