1*2+2*3+3*4+4*5+…+n(n+1)(n为正整数)求式子的结果!
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![1*2+2*3+3*4+4*5+…+n(n+1)(n为正整数)求式子的结果!](/uploads/image/z/2478504-48-4.jpg?t=1%2A2%2B2%2A3%2B3%2A4%2B4%2A5%2B%E2%80%A6%2Bn%28n%2B1%29%28n%E4%B8%BA%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%89%E6%B1%82%E5%BC%8F%E5%AD%90%E7%9A%84%E7%BB%93%E6%9E%9C%EF%BC%81)
1*2+2*3+3*4+4*5+…+n(n+1)(n为正整数)求式子的结果!
1*2+2*3+3*4+4*5+…+n(n+1)(n为正整数)
求式子的结果!
1*2+2*3+3*4+4*5+…+n(n+1)(n为正整数)求式子的结果!
因为:1×2=1/3×1×2×3
1×2+2×3=1/3×2×3×4
1×2+2×3+3×4=1/3×3×4×5
1×2+2×3+3×4+4×5=1/3×4×5×6,.
结论:1×2+2×3+3×4+…+n(n+1)= 1/3n(n+1)(n+2)
证明
原式=1/2n(n+1)+1/6n(n+1)(2n+1)
=1/6n(n+1)(2n+4)
=1/3n(n+1)(n+2)
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1*2+2*3+3*4+4*5+…+n(n+1)
=1/3 [1*2*(3-0)+2*3*(4-1)+3*4*(5-2)+....+n(n+1)*[(n+2)-(n-1)]]
=1/3[1*2*3-0*1*2+2*3*4-1*2*3+3*4*5-2*3*4+....+n*(n+1)(n+2)-(n-1)*n*(n+1)]
=1/3 [n*(n+1)(n+2)]