There seems to be an error: 24 should read 25.
| n |
1 |
2 |
3 |
4 |
5 |
6 |
| x |
1 |
5 |
13 |
25 |
41 |
61 |
| y |
4 |
8 |
12 |
16 |
20 |
24 |
We can work out x and y in terms of n, which just counts the data.
It’s clear that y=4n, but what is x? We can find out by using an iterative formula.
x[n+1]=x[n]+4n where square brackets mean subscript.
x₁=1.
So x₂=x₁+4=5, x₃=x₂+2×4=13, x₄=x₃+3×4=25, etc.
We can write these in terms of x₁:
x₂=x₁+4, x₃=x₁+4+8, x₄=x₁+4+8+12, etc.
In general x[n]=1+4∑i {1,n-1}=1+4n(n-1)/2, because the sum of the first n natural numbers is n(n+1)/2.
So x=1+2n(n-1)=2n²-2n+1. But we want to relate x and y, so we express n in terms of y: y=4n, so n=y/4 and we substitute this into the equation for x: x=2(y/4)²-2(y/4)+1=y²/8-y/2+1.
This can be written x=⅛(y²-4y+8), which is a sideways parabola.