The sum of the first four terms in the expansion of (1-2x)4 is equal to the sum of the first three terms in the expansion of (1-x)8. Show that this is true only if x=0 or x=- rac14.

9. The sum of the first four terms in the expansion of $(1-2x)^4$ is equal to the sum of the first three terms in the expansion of $(1-x)^8$. Show that this is true only if $x=0$ or $x=-\frac{1}{4}$.

Step 1: Find the sum of the first four terms of $(1-2x)^4$.
Using the binomial expansion $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$, with $a=1$, $b=-2x$, and $n=4$:
The first four terms are for $k=0, 1, 2,