Question

观察下列各式:1×2×3×4+1=52=(12+3×1+1)2, 2×3×4×5...

观察下列各式:1×2×3×4+1=52=(12+3×1+1)2, 2×3×4×5+1=112=(22+3×2+1)2, 3×4×5×6+1=192=(32+3×3+1)2, 4×5×6×7+1=292=(42+3×4+1)2, … (1)根据你观察、归纳、发现的规律,写出8×9×10×11+1的结果; (2)试猜想:n(n+1)(n+2)(n+3)+1是哪一个数的平方?并说明理由.

Answer 1

(1)观察下列各式:1×2×3×4+1=52=(12+3×1+1)2,2×3×4×5+1=112=(22+3×2+1)2,
3×4×5×6+1=192=(32+3×3+1)2,4×5×6×7+1=292=(42+3×4+1)2,得出规律:n(n+1)(n+2)(n+3)+1=(n2+3×n+1)2(n≥1),
8×9×10×11+1=(82+3×8+1)2=892;
(2)根据(1)得出的结论得出:
n(n+1)(n+2)(n+3)+1
=n(n+3)(n+1)(n+2)+1
=(n2+3n)(n2+3n+2)+1
=(n2+3n)2+2(n2+3n)+1
=(n2+3n+1)2.