Question

定義映射f:A→B,其中A={(m,n)|m,n∈R},B=R,已知對(duì)所有的有序正整數(shù)對(duì)(m,n)滿足下述條件:
①f(m,1)=1,②若n>m,f(m,n)=0;③f(m+1,n)=n[f(m,n)+f(m,n-1)],則f(2,2)=　　　　;f(n,2)=　　　　.

Answer 1

2　2ⁿ-2

根據(jù)已知得,f(1,2)=0=2¹-2,
f(2,2)=f(1+1,2)=2[f(1,2)+f(1,1)]=2f(1,1)
=2×1=2,
f(3,2)=f(2+1,2)=2[f(2,2)+f(2,1)]=2×(2+1)
=6=2³-2,
f(4,2)=f(3+1,2)=2[f(3,2)+f(3,1)]=2×(6+1)
=14=2⁴-2,
f(5,2)=f(4+1,2)=2[f(4,2)+f(4,1)]=2×(14+1)
=30=2⁵-2,
所以根據(jù)歸納推理可知f(n,2)=2ⁿ-2.