Ma puteti ajuta va rog?
[tex] \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + ... + \frac{1}{x(x + 1)} = \frac{2018}{2019} [/tex]
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Răspuns:

x = 2018.

Explicație pas cu pas:

1 / 1*2  +  1 / 2*3 + 1 / 3*4 + 1/ 4*5 + 1 / 5*6 + ... + 1 / x(x+1) =

1/1 - 1/2 +

1/2 - 1/3 +

1/3 - 1/4 +

1/4 - 1/5 +

1/5 - 1/6 +

- - - - - - - -

1/x - 1/(x+1) = unde se observa reducerile termenilor pe directii diagonale si ne ramane:

1 - 1(x+1).

1 - 1/(x+1) = 2018/2019

(x+1-1)/(x+1) = 2018/2019

x/(x+1) = 2018/2019

2019x = 2018(x+1)

2019x = 2018x + 2018

2019x - 2018x = 2018

x = 2018.