1. Zadatak

Opći se član može napisati u obliku $$a_n=\left(1+\frac1n\right)\left(1-\frac1{n^2}\right)=\left(1-\frac1n\right)\left(1+\frac1n\right)^2=\frac{(n-1)(n+1)^2}{n^3},$$ pa je $$\begin{aligned} P_k&=a_2\cdot a_3\cdot\ldots\cdot a_k\\ &=\frac{1\cdot3^2}{2^3}\cdot\frac{2\cdot4^2}{3^3}\cdot\frac{3\cdot5^2}{4^3}\cdot\ldots\cdot\frac{(k-1)(k+1)^2}{k^3}\\ &=\frac1{2^2}\cdot1\cdot1\cdot\ldots\cdot1\cdot\frac{(k+1)^2}k\\ &=\frac{(k+1)^2}{4k}. \end{aligned}$$ Dalje je $$\begin{aligned} \frac{(k+1)^2}{4k}>1000&\quad\Longleftrightarrow\quad k^2-3998k+1>0\\ &\quad\Longleftrightarrow\quad k^2-3998k>-1\quad\Longleftrightarrow\quad k(k-3998)>-1, \end{aligned}$$ Jer je $k>0$, mora biti $k-3998\ge0$.

Dakle, $k\ge3998$, $k\in\mathbb N$.

Najmanji takav $k=3998$.