Question

1×2×3×4+1=25=5² 2×3×4×5+1=121=11² 3×4×5×6+1=361=19²

1×2×3×4+1=25=5²
2×3×4×5+1=121=11²
3×4×5×6+1=361=19²
你能得出什么结论？请说明其正确性。（说明正确性就好了，结论知道）

Answer 1

(n-1)*n*(n+1)*(n+2)+1
=n(n²-1)*(n+2)+1
=n(n³-n+2n²-2)+1
=n^4-n²+2n³-2n+1
=n^4+2n³+n²-2n²-2n+1
=n²(n+1)²-2n(n+1)+1
=[n(n+1)-1]²
=(n²+n-1)²

追问

请说明其正确性！

追答

根据题中的数据规律得出
(n-1)*n*(n+1)*(n+2)+1
=n(n²-1)*(n+2)+1
=n(n³-n+2n²-2)+1
=n^4-n²+2n³-2n+1
=n^4+2n³+n²-2n²-2n+1
=n²(n+1)²-2n(n+1)+1
=[n(n+1)-1]²
=(n²+n-1)²
其中n>1
以上是推导步骤
可将数据带入以验证正确性
带入2得
1×2×3×4+1=5²
带入3得
2×3×4×5+1=11²
验证正确！