Đáp án + Giải thích các bước giải:
`x - 20/(11*13) - 20/(13*15) - 20/(15*17)-...-20/(53*55)=3/11` $\\$ `=> x - (20/(11*13) + 20/(13*15)+20/(15*17)+...+20/(53*55))=3/11` $\\$ `=>x - [10(2/(11*13)+2/(13*15) + 2/(15*17) + ... + 2/(53*55))]=3/11` $\\$ `=>x - [10*(1/11 - 1/13 + 1/13 - 1/15 + ... + 1/53 - 1/55)]=3/11` $\\$ `=> x - [10*(1/11 - 1/55)] = 3/11` $\\$ `=> x - (10*4/55) = 3/11` $\\$ `=> x - 8/11 = 3/11 => x = 3/11 + 8/11 = 11/11 = 1`
Vậy `x = 1`
`1/3 + 1/6 + 1/10 +... + 2/[x(x + 1)] = 2003/2004` $\\$ `=> 2/6+2/12+2/20+...+2/[x(x+1)] = 2003/2004` $\\$ `=> 2/(2*3) + 2/(3*4) + 2/(4*5) + ... + 2/[x(x+1)]=2003/2004` $\\$ `=> 2(1/(2*3) + 1/(3*4) + 1/(4*5) + ... + 1/[x(x + 1)]) = 2003/2004` $\\$ `=> 2(1/2 - 1/3 + ... + 1/x - 1/(x + 1)) = 2003/2004` $\\$ `=> 2(1/2 - 1/(x + 1)) = 2003/2004` $\\$ `=> 1/2 - 1/(x + 1) = 2003/4008` $\\$ `=> 1/(x + 1) = 1/2 - 2003/4008 = 1/4008` $\\$ `=> x + 1 = 4008 => x = 4007`
Vậy `x = 4007`
