题目

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 9
根据以上结果猜想（n+1)(n+2)(n+3)(n+4)+1=( )²

提问时间：2020-10-20

答案

{n+1}{n+2}{n+3}{n+4}+1
=[{n+1}{n+4}][{n+2}{n+3}]+1
=[{n^2+5n+4}][{n^2+5n+6}]+1
=[{n^2+5n+4}][{n^2+5n+4+2}]+1
=[{n^2+5n+4}]^2+2*[{n^2+5n+4}]*1+1
=[{n^2+5n+4}+1]^2
={n^2+5n+5}^2