阅读以下求1+2+3+…+n的值的过程:因为(n+1)²-n²=2n+1,n²-(n-1)²=2(n-1)+1,……2²-1²=2×1+1以上各式相加得(n+1)²-1=2(1+2+…+n)+n,所以1+2+3+…+n=(n²+2n-n)/2=n(n+1)/2.类
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![阅读以下求1+2+3+…+n的值的过程:因为(n+1)²-n²=2n+1,n²-(n-1)²=2(n-1)+1,……2²-1²=2×1+1以上各式相加得(n+1)²-1=2(1+2+…+n)+n,所以1+2+3+…+n=(n²+2n-n)/2=n(n+1)/2.类](/uploads/image/z/5477270-14-0.jpg?t=%E9%98%85%E8%AF%BB%E4%BB%A5%E4%B8%8B%E6%B1%821%2B2%2B3%2B%E2%80%A6%2Bn%E7%9A%84%E5%80%BC%E7%9A%84%E8%BF%87%E7%A8%8B%EF%BC%9A%E5%9B%A0%E4%B8%BA%EF%BC%88n%2B1%EF%BC%89%26sup2%3B-n%26sup2%3B%3D2n%2B1%2Cn%26sup2%3B-%EF%BC%88n-1%EF%BC%89%26sup2%3B%3D2%EF%BC%88n-1%EF%BC%89%2B1%2C%E2%80%A6%E2%80%A62%26sup2%3B-1%26sup2%3B%3D2%C3%971%2B1%E4%BB%A5%E4%B8%8A%E5%90%84%E5%BC%8F%E7%9B%B8%E5%8A%A0%E5%BE%97%EF%BC%88n%2B1%EF%BC%89%26sup2%3B-1%3D2%EF%BC%881%2B2%2B%E2%80%A6%2Bn%EF%BC%89%2Bn%2C%E6%89%80%E4%BB%A51%2B2%2B3%2B%E2%80%A6%2Bn%3D%28n%26sup2%3B%2B2n-n%29%2F2%3Dn%28n%2B1%29%2F2.%E7%B1%BB)
阅读以下求1+2+3+…+n的值的过程:因为(n+1)²-n²=2n+1,n²-(n-1)²=2(n-1)+1,……2²-1²=2×1+1以上各式相加得(n+1)²-1=2(1+2+…+n)+n,所以1+2+3+…+n=(n²+2n-n)/2=n(n+1)/2.类
阅读以下求1+2+3+…+n的值的过程:
因为(n+1)²-n²=2n+1,
n²-(n-1)²=2(n-1)+1,
……
2²-1²=2×1+1
以上各式相加得(n+1)²-1=2(1+2+…+n)+n,
所以1+2+3+…+n=(n²+2n-n)/2=n(n+1)/2.
类比以上过程,求1²+2²+3²+…+n²的值
阅读以下求1+2+3+…+n的值的过程:因为(n+1)²-n²=2n+1,n²-(n-1)²=2(n-1)+1,……2²-1²=2×1+1以上各式相加得(n+1)²-1=2(1+2+…+n)+n,所以1+2+3+…+n=(n²+2n-n)/2=n(n+1)/2.类
(n+1)³-n³=3n²+3n+1
n³-(n-1)³=3(n-1)²+3(n-1)+1
……
2³-1³=3×1²+3×1+1
相加
(n+1)³-1³=3(1²+2²+……+n²)+3(1+2+……+n)+n
1+2+……+n=n(n+1)/2
所以1²+2²+……+n²=n(n+1)(2n+1)/6
(n+1)^3-n^3=3n^2+3n+1,
n^3-(n-1)^3=3(n-1)^2+3(n-1)+1,
……
2^3-1^3=3*1^2+3*1+1
以上各式相加得
(n+1)^3-1^3=3(1²+2²+3²+…+n²)+3(1+2+……+n)+n
把上面的结果带入就出来了