Question

观察下列运算并填空.1*2*3*4+1=25=5的平方；2*3*4*5+1=121=11的平方：3*4*5*6+1=36

Answer 1

规律：a×(a+1)×(a+2)×(a+3)+1=[a×(a+3)+1]^2即四个连续递增的正整数的积加1等于第一个数乘以第四个数加上1的和的平方证：[a×(a+3)+1]^=(a^2+3a+1)^2=a^4+(3a+1)^2+2a^2*(3a+1)=a^4+6a^3+11a^2+6a+1a×(a+1)×(a+2)×(a+3)+1=(a^2+a)×(a^2+5a+6)+1=a^4+5a^3+6a^2+a^3+5a^2+6a+1=a^4+6a^3+11a^2+6a+1或者这样证：（为方便输入,以N代替A）n(n+1)(n+2)(n+3)+1=(n^2+3n+1)^2n(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