數列求和快捷公式?小吶不帥但很實在2021-03-16 17:30:39

)1+2+3+……+n=n(n+1)÷2

2)1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)÷6

3) 1^3+2^3+3^3+……+n^3=( 1+2+3+……+n)^2

=n^2*(n+1)^2÷4

4) 1*2+2*3+3*4+……+n(n+1)

=n(n+1)(n+2)÷3

5) 1*2*3+2*3*4+3*4*5+……+n(n+1)(n+2)

=n(n+1)(n+2)(n+3)÷4

6) 1+3+6+10+15+……

=1+(1+2)+(1+2+3)+(1+2+3+4)+……+(1+2+3+。。。+n)

=[1*2+2*3+3*4+……+n(n+1)]/2=n(n+1)(n+2) ÷6

7)1+2+4+7+11+……

=1+(1+1)+(1+1+2)+(1+1+2+3)+……+(1+1+2+3+。。。+n)

=(n+1)*1+[1*2+2*3+3*4+……+n(n+1)]/2

=(n+1)+n(n+1)(n+2) ÷6

8)1/2+1/2*3+1/3*4+……+1/n(n+1)

=1-1/(n+1)=n÷(n+1)

9)1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+……+1/(1+2+3+。。。+n)

=2/2*3+2/3*4+2/4*5+……+2/n(n+1)

=(n-1) ÷(n+1)

10)1/1*2+2/2*3+3/2*3*4+……+(n-1)/2*3*4*。。。*n

=(2*3*4*。。。*n- 1)/2*3*4*。。。*n

11)1^2+3^2+5^2+……。。。。(2n-1)^2=n(4n^2-1) ÷3

12)1^3+3^3+5^3+……。。。。(2n-1)^3=n^2(2n^2-1)

13)1^4+2^4+3^4+……。。。。+n^4

=n(n+1)(2n+1)(3n^2+3n-1) ÷30

14)1^5+2^5+3^5+……。。。。+n^5

=n^2 (n+1)^2