Question

If on day 1 you have 1 tile and day 2, 5 tiles and day 3, 13 tiles how many will you have on day 10 and day 50

Answer 1

This is a question of modeling data.
From that perspective, you’re given three data points and asked to extrapolate the data.

It’s likely that you’re looking for a polynomial model for the data.
Since you’re given three points, that suggests looking for a quadratic model,
In other words, finding a, b, and c, so that f(x) = ax^2 + bx + c fits your data
for x = 1, 2, and 3.

We suspect it’s quadratic because three points determine a polynomial
in the same way two points determine a linear polynomial (a line).

So with that in mind, we want to find a, b, and c.
We know that f(1) = 1, f(2) = 5 and f(3) = 13.
We know f(x) = ax^2 + bx + c for values x = 1, 2, and 3,
So 1*a + 1*b + 1*c = 1, 4*a + 2*b + c = 5, 9*a + 4*b + c = 13.
This gives us three equations. And we want to solve for the three unknowns a, b, and c.

You just want to solve that system of equations to find a, b, and c.

Answer 2

Then you have found explicitly what quadratic equation models your data, and then you can plug x = 10 and x = 50 in to find what's happening at days 10 and 50.