How To Do Citations In A Research Paper Apa Style Pascal’s Triangle and Cube Numbers

You are searching about How To Do Citations In A Research Paper Apa Style, today we will share with you article about How To Do Citations In A Research Paper Apa Style was compiled and edited by our team from many sources on the internet. Hope this article on the topic How To Do Citations In A Research Paper Apa Style is useful to you.

Pascal’s Triangle and Cube Numbers

To help explain where cubic numbers can be found in Pascal’s triangle, I will first briefly explain how square numbers are formed. The third diagonal of Pascal’s triangle is 1,3,6,10,15,21… If we add each of these numbers to its previous number, we get 0+1=1, 1+3=4, 3 +6= 9, 6+10=16… , which are square numbers. The way in which cubic numbers can be formed from Pascal’s triangle is similar, but a little more complex. While square numbers can be found on the third diagonal, for cubic numbers we must look at the fourth diagonal. The first rows of Pascal’s triangle are shown below, with these numbers in bold:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1615 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

This sequence is the tetrahedral numbers, the differences of which give the triangle numbers 1,3,6,10,15,21 (the sums of whole numbers, for example, 21 = 1+2+3+4+5). However, if you try to add consecutive pairs in the sequence 1,4,10,20,35,56, you will not get the numbers in the cube. To see how to obtain this sequence, we will have to look at the formula for tetrahedral numbers, which is (n)(n+1)(n+2)/6. If you expand this, you get (n^3 + 3n^2 + 2n)/6. Basically, we’re trying to do n^3, so a good starting point is that here we have a term^3/6, so we’re likely going to have to add. six tetrahedral numbers to make n^3, not 2. Try finding the cubic numbers from this information. If you are still stuck, see the next paragraph.

List the tetrahedral numbers with two zeros first: 0,0,1,4,10,20,35,56…

Then add three consecutive numbers at once, but multiply the middle one by 4:

0 + 0 x 4 + 1 = 1 = 1^3

0 + 1 x 4 + 4 = 8 = 2^3

1 + 4 x 4 + 10 = 27 = 3^3

4 + 10 x 4 + 20 = 64 = 4^3

10 + 20 x 4 + 35 = 125 = 5^3

This pattern, in fact, always continues. If you want to see why this is so, try expanding and simplifying (n(n+1)(n+2))/6 + 4(n-1)(n)(n+1)/6 + ( (n-2 )(n-1)n)/6, which are the formulas for the nth tetrahedral numbers, (n-1) and (n-2) and you should end up with n^3. If not, as I hope is the case (and I don’t blame you), just enjoy this interesting result and test it with your friends and family to see if they can spot this hidden link between Pascal’s triangle and cubic numbers.

Video about How To Do Citations In A Research Paper Apa Style

You can see more content about How To Do Citations In A Research Paper Apa Style on our youtube channel: Click Here

Question about How To Do Citations In A Research Paper Apa Style

If you have any questions about How To Do Citations In A Research Paper Apa Style, please let us know, all your questions or suggestions will help us improve in the following articles!

The article How To Do Citations In A Research Paper Apa Style was compiled by me and my team from many sources. If you find the article How To Do Citations In A Research Paper Apa Style helpful to you, please support the team Like or Share!

Rate Articles How To Do Citations In A Research Paper Apa Style

Rate: 4-5 stars
Ratings: 9085
Views: 15065857

Search keywords How To Do Citations In A Research Paper Apa Style

How To Do Citations In A Research Paper Apa Style
way How To Do Citations In A Research Paper Apa Style
tutorial How To Do Citations In A Research Paper Apa Style
How To Do Citations In A Research Paper Apa Style free
#Pascals #Triangle #Cube #Numbers

Source: https://ezinearticles.com/?Pascals-Triangle-and-Cube-Numbers&id=7320834