Question

2．（x+2y）⁷的展開(kāi)式中系數(shù)最大的項(xiàng)等于672x²y⁵．

Answer 1

分析 設(shè)r+1項(xiàng)系數(shù)最大，則有$\left\{\begin{array}{l}{{C}_{7}^{r}•{2}^{r}≥{C}_{7}^{r-1}•{2}^{r-1}}\\{{C}_{7}^{r}•{2}^{r}≥{C}_{7}^{r+1}•{2}^{r+1}}\end{array}\right.$，求出r，即可求出系數(shù)最大項(xiàng)．

解答 解：設(shè)r+1項(xiàng)系數(shù)最大，則有$\left\{\begin{array}{l}{{C}_{7}^{r}•{2}^{r}≥{C}_{7}^{r-1}•{2}^{r-1}}\\{{C}_{7}^{r}•{2}^{r}≥{C}_{7}^{r+1}•{2}^{r+1}}\end{array}\right.$，
_解得$\frac{13}{3}≤r≤\frac{16}{3}$
又∵0≤r≤7，∴r=5．
∴系數(shù)最大項(xiàng)為T(mén)₆=${C}_{7}^{5}$x²•2⁵y⁵=672x²y⁵．
故答案為：672x²y⁵．

點(diǎn)評(píng) 本題考查系數(shù)最大項(xiàng)，考查學(xué)生的計(jì)算能力，比較基礎(chǔ)．