观察1乘以2乘以3乘以4+1=25=5² 2乘以3乘以4乘以5+1=121=11²

题目

观察1乘以2乘以3乘以4+1=25=5² 2乘以3乘以4乘以5+1=121=11²
3乘以4乘以5乘以6+1=361=19²
（1）计算5乘以6乘以7乘以8+1的结果是多少?
（2）由此你能推出一般的结论吗?

答案

5*6*7*8+1=1681=41^2
n(n+1)(n+2)(n+3)+1
=[n(n+3)][(n+1)(n+2)]+1
=(n^2+3n)[(n^2+3n)+2]+1
=(n^2+3n)^2+2(n^2+3n)+1
=(n^2+3n+1)^2