- Problem Statement
- I was analyzing an algorithm for my datastructures/alg class, and I came upon the following numbers. For n = 1,2,3,4,5,6.... => 1,6,18,40,75,126.... respectively

- Relevant Equations
- (n^3+n^2)/2

The equation listed is the implicit form. I achieved this a weird way.

[itex]\begin{array}{|c|c|c|}

\hline 1 & 1 & 1*1 \\

\hline 2 & 6 & 2*3 \\

\hline 3 & 18 & 3*6 \\

\hline 4 & 40 & 4*10 \\

\hline 5 & 75 & 5 * 15 \\

\hline 6 & 126 & 6 * 21\\

\end{array}[/itex]

Focusing on the third column now, I can see as we go down the rows we have the left part of the product is =n, and the right part of the product I noticed can be expressed implicitly as well. So I have n* something.

I took at row 2 column three and though.. hmm, 3 = (2*3)/2, 6=(3*4)/2, 10= (4*5)/2 and I noticed the pattern ( (n)(n+1) ) /2. Multiply that with the n and I got my answer that it was O(n^4)

The thing is... sure I achieved the right answer but I just hate this process. It's like, you have to get lucky just for your brain to be able to figure out how to find the pattern. There isn't a sure way to do things, and if you can't find it under pressure on the test, you're screwed. There is no way to get the exact answer for any pattern, like there isn't step 1: do this, step 2: do this, like for example, going from ax^2 + bx+ c to y=a(x-h)^2 +k, cookie cutter method.

My question is: maybe there is a cookie cutter method for finding any pattern? Is there any more efficient way to do these kinds of problems? Not going to lie they are kind of fun to do but at the same time they are exhausting because there isn't a cookie cutter way of just getting the answer.

[itex]\begin{array}{|c|c|c|}

\hline 1 & 1 & 1*1 \\

\hline 2 & 6 & 2*3 \\

\hline 3 & 18 & 3*6 \\

\hline 4 & 40 & 4*10 \\

\hline 5 & 75 & 5 * 15 \\

\hline 6 & 126 & 6 * 21\\

\end{array}[/itex]

Focusing on the third column now, I can see as we go down the rows we have the left part of the product is =n, and the right part of the product I noticed can be expressed implicitly as well. So I have n* something.

I took at row 2 column three and though.. hmm, 3 = (2*3)/2, 6=(3*4)/2, 10= (4*5)/2 and I noticed the pattern ( (n)(n+1) ) /2. Multiply that with the n and I got my answer that it was O(n^4)

The thing is... sure I achieved the right answer but I just hate this process. It's like, you have to get lucky just for your brain to be able to figure out how to find the pattern. There isn't a sure way to do things, and if you can't find it under pressure on the test, you're screwed. There is no way to get the exact answer for any pattern, like there isn't step 1: do this, step 2: do this, like for example, going from ax^2 + bx+ c to y=a(x-h)^2 +k, cookie cutter method.

My question is: maybe there is a cookie cutter method for finding any pattern? Is there any more efficient way to do these kinds of problems? Not going to lie they are kind of fun to do but at the same time they are exhausting because there isn't a cookie cutter way of just getting the answer.