Răspuns:
x = 100
Explicație pas cu pas:
[tex]1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + ... + \frac{1}{1 + 2 + 3 + ... + x} = \frac{200}{101} [/tex]
folosim formula:
[tex]1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2} [/tex]
[tex]1 + \frac{1}{ \frac{2 \times 3}{2} } + \frac{1}{ \frac{3 \times 4}{2} } + ... + \frac{1}{ \frac{x(x + 1)}{2} } = \frac{200}{101} [/tex]
[tex]1 + \frac{2}{2 \times 3} + \frac{2}{3 \times 4} + ... + \frac{2}{x(x + 1)} = \frac{200}{101} [/tex]
[tex]2( \frac{1}{2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ... + \frac{1}{x(x + 1)}) = \frac{200}{101} [/tex]
[tex](1 - \frac{1}{2}) + ( \frac{1}{2} - \frac{1}{3}) + ( \frac{1}{3} - \frac{1}{4}) + ... + ( \frac{1}{x} - \frac{1}{x + 1}) = \frac{100}{101} [/tex]
[tex]1 - \frac{1}{x + 1} = \frac{100}{101} [/tex]
[tex]\frac{x + 1 - 1}{x + 1} = \frac{100}{101} = > \frac{x}{x + 1} = \frac{100}{101} [/tex]
[tex]101x = 100x + 100 = > x = 100[/tex]
