观察下列运算并填空:1×2×3×4+1=25=52;2×3×4×5+1=121=112:3×4×5×6+1=361=19
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观察下列运算并填空:
1×2×3×4+1=25=52;
2×3×4×5+1=121=112:
3×4×5×6+1=361=192;…
根据以上结果,猜想并研究:(n+1)(n+2)(n+3)(n+4)+1=______.
1×2×3×4+1=25=52;
2×3×4×5+1=121=112:
3×4×5×6+1=361=192;…
根据以上结果,猜想并研究:(n+1)(n+2)(n+3)(n+4)+1=______.
![观察下列运算并填空:1×2×3×4+1=25=52;2×3×4×5+1=121=112:3×4×5×6+1=361=19](/uploads/image/z/3388806-54-6.jpg?t=%E8%A7%82%E5%AF%9F%E4%B8%8B%E5%88%97%E8%BF%90%E7%AE%97%E5%B9%B6%E5%A1%AB%E7%A9%BA%EF%BC%9A1%C3%972%C3%973%C3%974%2B1%3D25%3D52%EF%BC%9B2%C3%973%C3%974%C3%975%2B1%3D121%3D112%EF%BC%9A3%C3%974%C3%975%C3%976%2B1%3D361%3D19)
由1×2×3×4+1=25=52=(02+5×0+5)2;
2×3×4×5+1=121=112=(12+5×1+5)2;
3×4×5×6+1=361=192=(22+5×2+5)2,…
观察发现:(n+1)(n+2)(n+3)(n+4)+1=(n2+5n+5)2.
证明:等式左边=(n+1)(n+2)(n+3)(n+4)+1
=(n2+3n+2)(n2+7n+12)+1
=n4+7n3+12n2+3n3+21n2+36n+2n2+14n+25
=n4+10n3+35n2+50n+25
=n4+2n2(5n+5)+(5n+5)2
=(n2+5n+5)2=等式右边.
故答案为:(n2+5n+5)2
2×3×4×5+1=121=112=(12+5×1+5)2;
3×4×5×6+1=361=192=(22+5×2+5)2,…
观察发现:(n+1)(n+2)(n+3)(n+4)+1=(n2+5n+5)2.
证明:等式左边=(n+1)(n+2)(n+3)(n+4)+1
=(n2+3n+2)(n2+7n+12)+1
=n4+7n3+12n2+3n3+21n2+36n+2n2+14n+25
=n4+10n3+35n2+50n+25
=n4+2n2(5n+5)+(5n+5)2
=(n2+5n+5)2=等式右边.
故答案为:(n2+5n+5)2
观察下列运算并填空 1×2×3×4+1=25=5² ......
观察下列运算并填空:1×2×3×4+1=25=52;2×3×4×5+1=121=112:3×4×5×6+1=361=19
观察下列运算并填空 1×2×3×4+1=25=52 2×3×4×5+1=121=112 3×4×5×6+1=361=19
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