$f(x)=x+(1-x)x^2+(1-x)(1-x^2)x^3+....(1-x)(1-x^2)....(1-x^{n-1})x^n, (n \ge 4)$

Question

Let $f(x)=x+(1-x)x^2+(1-x)(1-x^2)x^3+....(1-x)(1-x^2)....(1-x^{n-1})x^n, (n \ge 4)$

then

1. $f(x)=-\prod^n_{r=1} (1-x^r)$
2. $f(x)=1-\prod ^n_{r=1}(1-x^r)$
3. $f'(x)=(1-f(x))\left(\sum^{n}_{r=1}\frac{rx^{r-1}}{(1-x^r)}\right)$
4. $f'(x)=f(x)\left(\sum^{n}_{r=1}\frac{rx^{r-1}}{(1-x^r)}\right)$

I really don't know where to start for this quesion.

Answer 1

Perhaps this will help:$f(0)=0$, $f(1) =1$, $f'(0)=1$. Now we do elimination:

1.st can be right since there we get $f(0)=1$.

2.st can be right since there we get $f(1)=0$.

4.th can be right since there we get $f'(0) = f(0)$

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Or, you can try answering the question for $n=4$ (or less).

Answer 2

Calling

$$ g_n(x) = \sum_{j=1}^n\prod_{k=1}^j(1-x^k) $$

we have

$$f_n(x) = g_n(x)-g_{n-1}(x) = -\prod_{k=1}^n(1-x^k)$$