Question

在(√x+2/x)∧n的二项式展开式中,若常数项为60,则n等于( )

Answer 1

(√x+2/x)^n 二项式的展开式的通项为：

C(i,n) * (√x)^(n-i) * (2/x)^i = C(i,n) * 2^i * x^(n/2 - i/2 -i)

常数项为60,则

n/2 - i/2 - i = 0

n = 3i

C(i,n) * 2^i = 60

C(i,3i) * 2^i = 60

若 i=1 则上式左边 = C(1,3) * 2 = 6 ≠ 右边

若 i=2 则上式左边 = C(2,6) * 4 = 60 = 右边

所以 n = 3i = 6