作业帮mathematic induction
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/04/29 14:51:38
mathematic induction
Prove,by mathematic induction
a)1*2+2*3+3*4+.+n(n+1)=1/3*n(n+1)(n+2)
b)hence or otherwise find the value of
1+(1+2)+(1+2+3)+...+(1+2+3+.+100)
Prove,by mathematic induction
a)1*2+2*3+3*4+.+n(n+1)=1/3*n(n+1)(n+2)
b)hence or otherwise find the value of
1+(1+2)+(1+2+3)+...+(1+2+3+.+100)
a)
first:1*2=1/3*1*2*3=2
2nd:if 1*2+2*3+3*4+.+n(n+1)=1/3*n(n+1)(n+2) then:
1*2+2*3+3*4+.+n(n+1)+(n+1)(n+2)=1/3*n(n+1)(n+2)+(n+1)(n+2)=(n+1)(n+2)(n/3+1)=1/3(n+1)(n+2)(n+3)
so:1*2+2*3+3*4+.+n(n+1)=1/3*n(n+1)(n+2)
b)1+(1+2)+(1+2+3)+...+(1+2+3+.+100)
=1*2/2+2*3/2+3*4/2+.+100(100+1)/2=1/3*100*101*102/2=171700
first:1*2=1/3*1*2*3=2
2nd:if 1*2+2*3+3*4+.+n(n+1)=1/3*n(n+1)(n+2) then:
1*2+2*3+3*4+.+n(n+1)+(n+1)(n+2)=1/3*n(n+1)(n+2)+(n+1)(n+2)=(n+1)(n+2)(n/3+1)=1/3(n+1)(n+2)(n+3)
so:1*2+2*3+3*4+.+n(n+1)=1/3*n(n+1)(n+2)
b)1+(1+2)+(1+2+3)+...+(1+2+3+.+100)
=1*2/2+2*3/2+3*4/2+.+100(100+1)/2=1/3*100*101*102/2=171700