Đặt A= \(1\cdot3^2+3\cdot5^2+5\cdot7^2+\cdots+73\cdot75^2\)
\(=3^2\left(3-2\right)+5^2\left(5-2\right)+\cdots+75^2\left(75-2\right)\)
\(=\left(3^3+5^3+\cdots+75^3\right)-2\left(3^2+5^2+\cdots+75^2\right)\)
Đặt \(B=3^3+5^3+\cdots+75^3\)
\(=\left(1^3+2^3+3^3+4^3+\cdots+76^3\right)-1^3-\left(2^3+4^3+\cdots+76^3\right)\)
\(=\left(1+2+\cdots+76\right)^2-1-2^3\left(1^3+2^3+\cdots+38^3\right)\)
\(=\left(76\cdot\frac{77}{2}\right)^2-1-8\cdot\left(1+2+\cdots+38\right)^2\)
\(=\left(38\cdot77\right)^2-1-8\cdot\left(38\cdot\frac{39}{2}\right)^2=2926^2-1-8\cdot\left(19\cdot39\right)^2\)
\(=2926^2-1-8\cdot741^2=4168827\)
Đặt \(C=3^2+5^2+\cdots+75^2\)
\(=\left(1^2+2^2+3^2+4^2+\cdots+76^2\right)-1^2-\left(2^2+4^2+\cdots+76^2\right)\)
\(=\frac{76\left(76+1\right)\left(2\cdot76+1\right)}{6}-1-2^2\left(1^2+2^2+\cdots+38^2\right)\)
\(=\frac{76\cdot77\cdot153}{6}-1-4\cdot\frac{38\left(38+1\right)\left(2\cdot38+1\right)}{6}\)
\(=38\cdot77\cdot51-1-4\cdot\frac{38\cdot39\cdot77}{6}=149226-1-\frac23\cdot38\cdot39\cdot77\)
=149225-38*26*77
=73149
A=B-2C
=4168827-2*73149
=4022529
