Analysis of Algorithm - Part 3 of 3: Analytical solution

You will now find an analytical solution for hn. Your final formulas should not include any or operators.

To find an analytical solution for hn, it is useful to break down the work into two parts:

1 - Based on the first 10 terms of hn, guess non-recursive formulas for the even and odd terms of the sequence, i.e. one formula for hn when n is even and another formula when n is odd.

Hint: somewhere along the way, your two formulas will probably include a well known series or product that you can solve. You may have already done this in the previous page , but if not, now is the time.

Remember, iteration is a slow process where the journey usually reveals more information than the destination. This is definitely the case with this problem. If you do not see patterns in your answers, it is probably because you skipped some intermediate steps. Go back to the previous page, and slow down

2 - Give a single analytical solution for hn, i.e. a single formula that will work for all values of n starting at 0. In this part, you will need to figure out how to merge the even and odd cases into a single mathematical formula. This can usually be accomplished by using some of the mod, floor and ceiling operators. Before you attempt to do so, be sure that your two separate formulas are correct by checking them against the values for 0h0 to 9h9 that you calculated in the previous page. Your final solution should be as simple as possible.