Question

a=1x1^2+2x2^3+3x4^2+......+98x99^2

Answer 1

Sửa đề: \(a=1\cdot2^2+2\cdot3^2+3\cdot4^2+\cdots+98\cdot99^2\)

\(=2^2\left(2-1\right)+3^2\left(3-1\right)+4^2\left(4-1\right)+\cdots+99^2\left(99-1\right)\)

\(=\left(2_{}^3+3^3+\cdots+99^3\right)-\left(2^2+3^2+\cdots+99^2\right)\)

\(=\left(1^3+2^3+3^3+\cdots+99^3\right)-\left(1^2+2^2+3^2+\cdots+99^2\right)\)

\(=\left(1+2+\cdots+99\right)^2-\frac{99\cdot\left(99+1\right)\left(2\cdot99+1\right)}{6}\)

\(=\left(99\cdot\frac{100}{2}\right)^2-\frac{99\cdot100\cdot199}{6}=\left(99\cdot50\right)^2-33\cdot50\cdot199\)

\(=4950^2-328350=24174150\)