Gọi `A = 1 xx 199  + 2 xx 198 + 3 xx 197 + ... + 198 xx 2 + 199 xx 1`

`A = 1 xx 199 - 0 xx 1 + 2 xx 199 - 1 xx 2 + 3 xx 199 - 2 xx 3 + ... + 199 xx 199 - 198 xx 199`

`A = (1 xx 199  + 2 xx 199 + 3 xx 199 + ... + 199 xx 199) - (0 xx 1 + 1 xx 2 + 2 xx 3 + ... + 198 xx 199)`

`A = 199(1 + 2 + 3 + ... + 199) - (0 xx 1 + 1 xx 2 + 2 xx 3 + ... + 198 xx 199)`

  Gọi `B = 199(1 + 2 + 3 + ... + 199)`

        `C = 0 xx 1 + 1 xx 2 + 2 xx 3 + ... + 198 xx 199`

`=> B = 199(1 + 2 + 3 + ... + 199)`

  `->` Số số hạng trong ngoặc: `(199 - 1) : 1 + 1 = 199`(số)

  `->` Tổng trong ngoặc: `(199 + 1) xx 199 : 2 = 19900`

`=> B = 199 xx 19900`

`=> B = 3960100`

`=> C = 0 xx 1 + 1 xx 2 + 2 xx 3 + ... + 198 xx 199`

`=> C = 1 xx 2 + 2 xx 3 + 3 xx 4 + ... + 198 xx 199`

`=> 3C = 1 xx 2 xx 3 + 2 xx 3 xx (4 - 1) + 3 xx 4 xx (5 - 2) + ... + 198 xx 199 xx (200 - 197)`

`=> 3C = 1 xx 2 xx 3 - 1 xx 2 xx 3 + 2 xx 3 xx 4 - 2 xx 3 xx 4 + 3 xx 4 xx 5 - ... -197 xx 198 xx 199 + 198 xx 199 xx 200`

`=> 3C = 198 xx 199 xx 200`

`=> 3C = 7880400`

`=> C = 2626800`

  `=> A = B - C`

  `=> A = 3960100 - 2626800`

  `=> A = 1333300`

        Vậy: `1 xx 199  + 2 xx 198 + 3 xx 197 + ... + 198 xx 2 + 199 xx 1 = 1333300`

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