由a(1)+2a(2)+3a(3)+……na(n)=n(n+1)(n+2)求出一些項(xiàng)觀察一下:
a(1)=6=3(1+1)
a(2)=9=3(2+1)
a(3)=12=3(3+1)
a(4)=15=3(4+1)
所以,猜測
a(n)=3(n+1)
證明
當(dāng)n=1時(shí)
a(1)=6=1*(1+1)*(1+2),正確
假設(shè)n=k時(shí)正確,即
a(1)+2a(2)+3a(3)+……ka(k)=k(k+1)(k+2)
當(dāng)n=k+1時(shí)
a(1)+2a(2)+3a(3)+……ka(k)+(k+1)a(k+1)=
=k(k+1)(k+2)+(k+1)*3(k+1+1)=
=k(k+1)(k+2)+(k+1)*3(k+2)=
=(k+1)(k+2)(k+3)=
=(k+1)[(k+1)+1][(k+1)+2]
正確!