8 use the following algorithm to answer the next sub...

8. Use the following algorithm to answer the next sub-questions. for i = 1..n do for j = 1..i do print “The proof is in the pudding. . .” (a) (4 points) Give a closed-form sum over number of times the phrase in the inner loop is issued. (b) (4 points) Give a closed-form summation for every value that j takes in the inner loop

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00:01 Okay, so the easiest way to do this is to write each for loop as a sum.

00:07 So we're going to write the first for loop.

00:12 So you have i to 1, i equals 1 to n minus 3.

00:17 So let's write it as sum from i equals 1 to n minus 3 and then sum from j is equal to 1 not 1, n minus 2i, n minus 2i to n plus 3i n minus 2i to n plus 3i.

00:44 Okay, and then how many how many things are in there? well, it's important to note that there are only three statements inside the for loop and no others.

01:02 So how many statements are inside the for loop? it's just three.

01:07 So the total number of statements so total number of statements is going to be this quantity.

01:19 Okay, which you can also write as 3 times i equals 1 to n minus 3 sum from j equals n minus 2i to n plus 3i of 1.

01:35 Okay.

01:35 So now first let's do the inner sum.

01:37 If you go from n minus 2i to n plus 3i how many ones is that? well, this is going to be just the difference plus 1.

01:48 So this is going to be 3i equals 1 to n minus 3.

01:55 And so what is the difference between the two? it's going to be 5i.

02:00 So it'll be 5i plus 1, right? because that's how many indices there are in between when you take that sum.

02:16 Okay, so you have 5i plus 1 and now we can use formulas.

02:23 So again, okay.

02:30 So again, let's split this up...

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