There is a problem I'm working that diverges quite a bit from the ramp up problems I had with induction. Anyway, here it is:
You have (mx+ny)$^g$ and g is an element in the natural numbers $\Bbb N$.
What will the coefficient of x$^h$y$^g$$^-$$^h$ is g when h $\in$ $\Bbb N$ and 0 $\le$ h $\le$ g
I'm struggling even at the base case, and I'm stuck on the approach you'd use to solve this. Can this be approached some way other than induction as well?
Base Case
- Just going with h=1,g=2 you get m$^2$x$^2$+2mn xy + n$^2$y$^2$ There the coefficient is 2, so an initial suspicion would be the coeff. is related to the exponent g. If you carry that forward for h=2, g=3, the coefficient is 3 and so on. When the two (g and h) are the same value, the coefficent seems to be the evaluated value of y because the exponent on y is 0 and so the value is 1. Does the base case need to be split up?
Inductive Hypothesis
- This will naturally depend on what the base case is. I would think I'd want to prove this for k+1, but I struggle with what that should look like or if it would be split because of cases.
I would start by reducing the problem - let $m = n = 1$, so that we're just expanding out $(x + y)^g$. Then, try expanding out a few more cases:
$$\begin{eqnarray}(x + y)^0 & = & 1 = 1 x^0 y^0 \\ (x + y)^1 & = & x + y \\ (x + y)^2 & = & x^2 + 2xy + y^2 \\ (x + y)^3 & = & x^3 + 3x^2 y + 3x y^2 + y^3 \\ (x + y)^4 & = & x^4 + 4x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4 \end{eqnarray}$$
So it looks like the coefficients go something like:
$\begin{eqnarray} &&&& 1 \\ &&& 1 && 1 \\ && 1 && 2 && 1 \\ & 1 && 3 && 3 && 1\\ 1 && 4 && 6 && 4 && 1 \end{eqnarray}$
Does this look like anything you're familiar with?