How To Find A Formula For The Nth Term Pascal’s Triangle and Cube Numbers

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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 the previous number, we get 0+1=1, 1+3=4, 3 +6=9, 6+10= 16… , are square numbers. How cube numbers can be formed in Pascal’s triangle is similar, but more complicated. While square numbers can be found on the third diagonal, for cube numbers, we must look to the fourth diagonal. The first few rows of Pascal’s triangle are shown below, with the numbers in bold:

1 1

121

133 1

1 4 6 4 1

1 5 10 10 5 1

1615 20 15 6 1

1 721 35 35 21 7 1

1828 56 70 56 28 8 1

This sequence is a tetrahedral number, the difference of which gives the triangular numbers 1,3,6,10,15,21 (total numbers eg 21 = 1+2+3+4+5). However, if you try to match consecutive pairs in the order 1,4,10,20,35,56, you don’t get cubed numbers. To see how to find this sequence, we need to look at the formula for tetrahedral numbers, which is (n)(n+1)(n+2)/6. If you multiply this, you get (n^3 + 3n^2 + 2n)/6. Basically, we’re trying to do un^3, so a good starting point is that here we have…3/6 times, so we might want to add them together. six tetrahedral numbers make n^3, not 2. Try to find cube numbers from this information. If you are still stuck, refer to 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 really, really goes on. If you want to see why this is so, then try multiplying 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) and (n-2) tetrahedral numbers, and you should end up with n^3. Otherwise, as I expect it to be (and I don’t blame you), enjoy this interesting result and test it on your friends and family to see if they can spot this hidden link between Pascal’s triangle and cube numbers!

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