Dynamic Programming

Our recursive algorithm for computing Fibonacci numbers may look elegant, but that doesn't mean it's efficient. For example, when computing the fourth term in the sequence, it calculates the value for both the second and third terms. Likewise, when calculating the value of the third term in the sequence, it calculates the value for the first and second terms. This isn't ideal, as the second term in the sequence was already being calculated in order to get the fourth term. Dynamic programming will help us to address this problem by ensuring you break down the problem into the appropriate subproblems, and never solve the same subproblem twice.

Exercise 52: Summing Integers

In this exercise, you write a sum_to_n function to sum integers up to n. You store the results in a dictionary, and the function will use the stored results to return the answer in fewer iterations. For example, if you already know the sum of integers up to 5 is 15, you should...