Question

Help

Answer 1

The coefficient of x^n in the expansion of
(1+x)(1-x)^n

Answer 2

@Kainui

Answer 3

So what do you think so far? I'll help you get there. Don't forget to consider pascal's triangle.

Answer 4

I tried to distribute it as
(1-x)^n x(1-x)^n

Answer 5

@Kainui

Answer 6

Ok so far so good. So if you just look at the terms of x^n in the (1-x)^n term it should be apparent what this is correct? It will just be (-1)^n since the coefficient is just -1 and x^n is the highest term so it doesn't "mix" with the other terms, it is always on top.

However this is not the case for the next term. We now need to look at the second set of terms.

Answer 7

okay

Answer 8

By "second set of terms" what I mean is, well here is a simple example: \[(1-x)^3 = 1-3x+3x^2-1x^3\] So here we have n=3. So we can see this is just like I described earlier, (-1)^n is on the x^3 term. However when we distribute x over this, our 3rd power term is the "second" one down. \[x(1-x)^3 = 1x-3x^2+3x^3-1x^4\]
So we have to add 3 from this term.

So overall we have \[(-1+3)x^3\] and our n=3 term is 2.

See if you can generalize this result by using the binomial theorem since it gives you a definite result for the second column down the side of Pascal's triangle.