Question

设n∈N^*，（2x+1）ⁿ的展开式各项系数之和为a_n，（3x+1）ⁿ展开式的二项式系数之和为b_n，则

lim

n→+∞

2an+3bn

an+1bn+1

=______．

Answer 1

令x=1由二项式定理可得a_n=3ⁿ，（3x+1）ⁿ展开式的二项式系数之和b_n=2ⁿ
∴

lim

n→∞

2an+3bn

an+1bn+1

=

lim

n→∞

2?3n+3?2n

3n+1+2n+1

=

lim

n→∞

2+3?( 2 3 )n

3+2?( 2 3 )n

=

2+3 lim n→∞ ( 2 3 )n

3+2 lim n→∞ ( 2 3 )n

=

2

3

故答案为

2

3