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=-(1)/(8)

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 $x=-\frac{1}{8}$.

• Step 1: Expand $(1-2x)^4$ and find the sum of its first four terms. Using the binomial theorem $(A+B)^n = \sum_{r=0}^{n} \binom{n}{r} A^{n-r} B^r$: For $(1-2x)^4$, $A=1$, $B=-2x$, $n=4$. The terms are