已知 $a,b$ 为正数，且 $\dfrac 1a+\dfrac 1b=1$，且 $n\in\mathbb N^{\ast}$，求证：$(a+b)^n-a^n-b^n\geqslant 2^{2n}-2^{n+1}$．
解析 根据题意，有\[\begin{split} LHS&=\sum_{k=1}^{n-1}\mathop{\rm C}\nolimits_n^ka^kb^{n-k}\\ &=\dfrac 12\sum_{k=1}^{n-1}\left(\mathop{\rm C}\nolimits_n^ka^kb^{n-k}+\mathop{\rm C}\nolimits_n^{n-k}a^{n-k}b^k\right)\\ &\geqslant \sum_{k=1}^{n-1}\sqrt{\mathop{\rm C}\nolimits_n^ka^kb^{n-k}\cdot \mathop{\rm C}\nolimits_n^{n-k}a^{n-k}b^k}\\ &=\sum_{k=1}^{n-1}\mathop{\rm C}\nolimits_n^k(ab)^{\frac n2}\\ &\geqslant 2^n\sum_{k=1}^{n-1}\mathop{\rm C}\nolimits_n^k\\ &=2^{2n}-2^{n+1},\end{split}\]命题得证．