1*2+2*3.+9*10= 1*2+2*3.2008*2009= 1*2+2*3+.+n(n+1)= 1*2*3+2*

1个回答

  • ∵ n(n+1)=n²+n

    ∴ 1*2+2*3+.+n(n+1)=(1²+2²+3²+.+n²)+(1+2+3+.+n)

    =n(n+1)(2n+1)/6+n(n+1)/2=n(n+1)(n+2)/3

    ∴ 1*2+2*3+.+9*10=9*10*11/3=330

    1*2+2*3+.+2008*2009=2008*2009*2010/3=2702828240

    ∵ n(n+1)(n+2)= n^3+3*n^2+2*n

    ∴ 1*2*3+2*3*4.+n(n+1)(n+2)=(1^3+2^3+.n^3)+3(1^2+2^2+.n^2)+2(1+2+.+n)

    =[n(n+1)/2]^2+3[n(n+1)(n+2)/6]+2[n(n+1)/2]

    =n(n+1)(n+2)(n+3)/4

    一般地 有1*2*...*m+2*3*.*(m+1)+.+n*(n+1)*.(n+m-1)

    =n(n+1)...(n+m)/(m+1) 请记住这一公式