1Ã1ï¼2Ã2ï¼3Ã3ï¼â¦â¦ï¼nÃn=n(nï¼1)(2nï¼1)/6
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(nï¼1)^3=n^3+3n^2+3n+1
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2^3=1^3+3Ã1^2+3Ã1+1
3^3=2^3+3Ã2^2+3Ã2+1
4^3=3^3+3Ã3^2+3Ã3+1
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(n+1)^3=n^3+3n^2+3n+1
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(n+1)^3=1+3(1^2+2^2+3^2+â¦â¦+n^2)+3(1+2+3+â¦â¦+n)+n
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çåå°±æ¯ä¸ä¸ªçå·®æ°å,å为n(1+n)÷2,äºæ¯
(n+1)^3=1+3(1^2+2^2+3^2+â¦â¦+n^2)+3n(n+1)÷2+n
æ以, 3(1^2+2^2+3^2+â¦â¦+n^2)= (n+1)^3ï¼3n(n+1)÷2ï¼(n+1)
=n^3+3n^2+3n+1ï¼3n^2/2ï¼3n/2ï¼nï¼1
=n^3+3/2n^2+n/2
æ以, 1^2+2^2+3^2+â¦â¦+n^2=1/3(n^3+3n^2/2+n/2)
=n(n+1)(2n+1)/6
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