Thay \(a=t\) nhé ,t nhìn trất's hơn
\(M=\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t+4\right)+1\)
\(M=\left[\left(t+1\right)\left(t+4\right)\right]\left[\left(t+2\right)\left(t+3\right)\right]+1\)
\(M=\left[t\left(t+4\right)+1\left(t+4\right)\right]\left[t\left(t+3\right)+2\left(t+3\right)\right]+1\)
\(M=\left(t^2+4t+t+4\right)\left(t^2+3t+2t+6\right)+1\)
\(M=\left(t^2+5t+4\right)\left(t^2+5t+6\right)+1\)
\(M=\left(t^2+5t+5-1\right)\left(t^2+5t+5+1\right)+1\)
\(M=\left(t^2+5t+5\right)^2-1+1\)
\(M=\left(t^2+5t+5\right)^2\)
Vì \(t\in Z\Rightarrow\left(t^2+5t+5\right)\in Z\)
Nên \(M\) là bình phương của 1 số nguyên (đpcm)
M = (a+1)(a+2)(a+3)(a+4) +1
= [(a+1)(a+4)].[(a+2)(a+3)] + 1
= (a2 + 5a + 4).(a2 + 5a + 6) + 1
= (a2 + 5a + 4).(a2 + 5a + 4 + 2) + 1
= (a2 + 5a + 4)2 + 2.(a2 + 5a + 4) + 1
= (a2 + 5a + 4 + 1)2 = (a2 + 5a + 5)2
