1047

在 ${\displaystyle (x-\sqrt{2}{)}^{2024}}$ 的二项展开式中，含 ${\displaystyle x}$ 的奇次幂的项之和为 ${\displaystyle S}$, 当 ${\displaystyle x=\sqrt{2}}$ 时, ${\displaystyle S=}$ _____.

把 ${\displaystyle S}$ 表示为关于 ${\displaystyle x}$ 的函数 ${\displaystyle S(x)}$, 然后再记偶次幂的项之和 ${\displaystyle T(x)}$, 则$$\begin{array}{r}S(x)+T(x)=(x-\sqrt{2}{)}^{2024}\Rightarrow \{\begin{array}{rl}& S(\sqrt{2})+T(\sqrt{2})=0,\\ & S(-\sqrt{2})+T(-\sqrt{2})=2^{3036}.\end{array}\end{array}$$
此外，显然有${\displaystyle S(-\sqrt{2})=-S(\sqrt{2}),T(-\sqrt{2})=T(\sqrt{2})}$, 因此解得${\displaystyle S=-2^{3035}}$.