\(K=1\cdot99^2+2\cdot98^2+\cdots+49\cdot51^2\)
\(=99^2\left(100-99\right)+98^2\left(100-98\right)+\cdots+51^2\left(100-51\right)\)
\(=100\left(51^2+52^2+\cdots+99^2+99^2\right)-\left(51^3+52^3+\cdots+98^3+99^3\right)\)
Đặt \(A=51^2+52^2+\cdots+99^2\)
\(=\left(1^2+2^2+\cdots+50^2+51^2+52^2+\cdots+99^2\right)-\left(1^2+2^2+\cdots+50^2\right)\)
\(=\frac{99\left(99+1\right)\left(2\cdot99+1\right)}{6}-\frac{50\left(50+1\right)\left(2\cdot50+1\right)}{6}\)
\(=\frac{99\cdot100\cdot199}{6}-\frac{50\cdot51\cdot101}{6}\)
\(=33\cdot50\cdot199-25\cdot17\cdot101\)
=285425
Đặt \(B=51^3+52^3+\cdots+99^3\)
\(=\left(1^3+2^3+\cdots+50^3+51^3+52^3+\cdots+99^3\right)-\left(1^3+2^3+\cdots+50^3\right)\)
\(=\left(1+2+\cdots+99\right)^2-\left(1+2+\cdots+50\right)^2\)
\(=\left\lbrack99\cdot\frac{100}{2}\right\rbrack^2-\left\lbrack50\cdot\frac{51}{2}\right\rbrack^2=\left(99\cdot50\right)^2-\left(25\cdot51\right)^2\)
\(=4950^2-1275^2=22876875\)
Ta có: \(K=100\left(51^2+52^2+\cdots+99^2+99^2\right)-\left(51^3+52^3+\cdots+98^3+99^3\right)\)
=100A-B
=28542500-22876875
=5665625
