Folosim formula lui Gauss:
[tex] \boxed{1+2+3+\ldots + n=\dfrac{n(n+1)}{2}} [/tex]
[tex] f= \dfrac{1+2+3+\ldots + 74}{75+76+\ldots + 999} \\ f=\dfrac{\dfrac{74\cdot 75}{2}}{\dfrac{999\cdot 1000}{2}-\dfrac{74\cdot 75}{2}} \\ f=\dfrac{74\cdot 75}{999\cdot 1000-74\cdot 75} \\ f= \dfrac{25\cdot 74\cdot 3}{25(999\cdot 40-74\cdot 3)} \\ f=\dfrac{3\cdot 74}{3(333\cdot 40-74)} \\ f=\dfrac{2\cdot 37}{2(333\cdot 20-37)} \\ f=\dfrac{37}{6660-47} \\ f=\dfrac{37}{6623} \Rightarrow \tt f=\dfrac{1}{179} [/tex]
