關於MI既一條問題

根據我查教科書﹐Tn是n(n+1)
令P(n)為命題"對任意正整數n,T1+T2+...Tn=1/3n(n+1 )(n+2)"即
"對任意正整數n,1*2+2*3+...+n(n+1)=1/3n(n+1 )(n+2)"
當n=1
LHS=1*2=2
RHS=1/3(1+1)(1+2)=2
LHS=RHS
所以當n=1﹐命題成立
設當n=k時﹐命題成立
T1+T2+...Tk=1/3k(k+1 )(k+2)
當n=k+1時
LHS
=T1+T2+...Tk+Tk+1
=1/3k(k+1 )(k+2)+Tk+1 (用假設)
=1/3k(k+1 )(k+2)+(k+1)(k+2)
=1/3(k+1 )(k+2)(k+3)
=1/3(k+1)(k+1+1)(k+1+2)
=RHS
所以當n=k+1時,命題成立
根據數學歸納法﹐對任意正整數n﹐P(n)成立