You are searching about *Find A Formula For The Nth Term Of The Sequence*, today we will share with you article about Find A Formula For The Nth Term Of The Sequence was compiled and edited by our team from many sources on the internet. Hope this article on the topic **Find A Formula For The Nth Term Of The Sequence** is useful to you.

Muc lục nội dung

## Pascal’s Triangle and Cube Numbers

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

1 6 15 **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, whose differences give the triangle numbers 1,3,6,10,15,21 (the sums of whole numbers e.g. 21 = 1+2+3+4+5). However, if you try adding up consecutive pairs in the sequence 1,4,10,20,35,56, you do not get the cube numbers. To see how to get 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, it you get (n^3 + 3n^2 + 2n)/6. Basically, we are trying to make n^3, so a good starting point is that here we have a n^3/6 term, so we are likely to need to add together *six* tetrahedral numbers to make n^3, not 2. Have a go at trying to find the cube numbers from this information. If you’re still stuck, then look at 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 a time, 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 does in fact, always continue. If you want to see why this is the case, then try exanding 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, (n-1)th and (n-2)th tetrahedral numbers, and you should end up with n^3. Otherwise, as I expect is the case (and I don’t blame you), just enjoy the this interesting result and test it out on your friends and family to find out if they can spot this hidden link between Pascal’s triangle and cube numbers!

## Video about Find A Formula For The Nth Term Of The Sequence

You can see more content about **Find A Formula For The Nth Term Of The Sequence** on our youtube channel: Click Here

## Question about Find A Formula For The Nth Term Of The Sequence

If you have any questions about **Find A Formula For The Nth Term Of The Sequence**, please let us know, all your questions or suggestions will help us improve in the following articles!

The article **Find A Formula For The Nth Term Of The Sequence** was compiled by me and my team from many sources. If you find the article Find A Formula For The Nth Term Of The Sequence helpful to you, please support the team Like or Share!

## Rate Articles Find A Formula For The Nth Term Of The Sequence

**Rate:** 4-5 stars

**Ratings:** 2924

**Views:** 54535083

## Search keywords Find A Formula For The Nth Term Of The Sequence

Find A Formula For The Nth Term Of The Sequence

way Find A Formula For The Nth Term Of The Sequence

tutorial Find A Formula For The Nth Term Of The Sequence

Find A Formula For The Nth Term Of The Sequence free

#Pascals #Triangle #Cube #Numbers

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