1x2÷1+2x3÷1+3x4÷1+4x5÷1+5x6÷1.+n(n+1)÷1

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1x2÷1+2x3÷1+3x4÷1+4x5÷1+5x6÷1.+n(n+1)÷11x2÷1+2x3÷1+3x4÷1+4x5÷1+5x6÷1.+n(n+1)÷11x2÷1+2x3÷1+3x4÷1+4x5÷

1x2÷1+2x3÷1+3x4÷1+4x5÷1+5x6÷1.+n(n+1)÷1
1x2÷1+2x3÷1+3x4÷1+4x5÷1+5x6÷1.+n(n+1)÷1

1x2÷1+2x3÷1+3x4÷1+4x5÷1+5x6÷1.+n(n+1)÷1
原式=1*2+2*3+3*4+...+n(n+1)
=1²+1+2²+2+3²+3+...+n²+n
=(1²+2²+3²+...+n²)+(1+2+3+...+n)
∵1²+2²+3²+...+n²=【n(n+1)(2n+1)】/6①(平方和公式)
1+2+3+...+n=【n(n+1)】/2②(等差数列公式)
∴原式=①+②=【n(n+1)(n+2)】/3