In the picture of Pascal’s triangle shown, what type of numbers appear in the diagonal indicated by the arrow?
This is the diagonal we’re interested in. The first three values are one, three, and then six. Let’s see if we can get a few more values. In Pascal’s triangle, an element is found by taking the sum of the two elements above that space. Above the blank space here is a one and a three. And so the missing element is a four. On the next row, we have a space underneath the one and four, which should be a five. The space underneath the four and six equals 10. Again, we have a space under a six and a four, which will be 10, and a four and one, which will be five. We’ve now found a fourth element on our diagonal, which is a 10. And we can determine that the fifth element would be a 15.
Let’s see if this is enough information for us to determine what type of numbers one, three, six, 10, and 15 are. To do this, let’s draw some symbols. For the one, we have one dot. To get from one to three, we need to add two dots. We have two new dots and the one original. What about going from three to six? We need three new dots. We have three new dots plus the two from the last stage plus the one that we started with. From six to 10, we’re adding four new. So we began with a row of four. We’ll stack our row of three and then two with the final one. Now, we see a pattern. To go from 10 to 15, we add five. The new bottom row will have five. To that five, we’ll stack the 10 from the previous iteration. This shape clues us in on what type of numbers this diagonal represents.
The numbers one, three, six, and 10 continuing across this diagonal we would call the triangular numbers.