Friday Tri-day! As is, Pascal’s Tri-day…
We found three sequences (four if you count 1, 1, 1, 1, 1, 1…) last week in Pascal’s triangle: counting numbers, triangular numbers and square numbers.
Today we will start by adding up the rows of the triangle:
If you’re a computer-lover you might recognise these as much of computing is built upon them (via binary numbers). They are the powers of 2. You might not know what this means, so I’ve made a table to explain:
|Written out||2/2||2||2 x 2||2 x 2 x 2||2 x 2 x 2 x 2||2 x 2 x 2 x 2 x 2|
So 23 means you write down three 2s and multiply them all together. The sequence is the powers of 2.
Now we’ll look at the sum of the shallow diagonals. Have a look:
The sequence we find when we add up along the shallow diagonals is 1, 1, 2, 3, 5, 8, 13 … This is the Fibonacci sequence of numbers, which you can continue by adding the two previous numbers together (1 + 1 = 2; 1 + 2 = 3; 2 + 3 = 5 etc). There are all sorts of connections to other areas of mathematics from the Fibonacci numbers, and also into nature.These include shells, flowers and leaves on plants. It’s amazing to find maths in nature when we so often consider it a human construct – it’s not!
Now if we look at the top few rows of Pascal’s triangle we see:
If you look at the second row you see 11; the third row says 121. 121 is 11 x 11; 1331 is 11 x 11 x 11. These are the powers of 11! The next row of Pascal’s triangle is 1 5 10 10 5 1. How can we find the power of 11 from this row? Here is the method, carrying tens over to the next column, which gives 11 to the power of 5.
From the picture you can see that the row 1, 6, 15, 20, 15, 6, 1 gives us 1771561, or 11 to the power of 6.
Pascal’s triangle contains not only the counting numbers, triangular numbers and square numbers, but also the powers of 2, powers of 11 and Fibonacci’s sequence. How amazing is that from something that starts off seeming so simple and uninspiring!
I have one more post coming up next week on Pascal’s triangle, and it’s a pretty one! So tune back in then!
See the rest of my Pascal’s Triangle posts at