数列求和常用公式: 1)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 (2n^2+2n-1) ÷ 12 15)1+2+2^2+2^3+.+2^n=2^(n+1) – 1 |
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