For which positive integers $n$ does the inequality

$$
(n-2)\left(2+2^{2}+\cdots+2^{n}\right)<n \cdot 2^{n}
$$

hold?
ekkor

The sum of the powers of 2 in parentheses can be written as the sum of a geometric sequence where $a_{1}=2, q=2$;

$$
S_{n}=2\left(2^{n}-1\right) \text {. }
$$

Thus, the inequality (1) can be written as:

$$
(n-2) 2\left(2^{n}-1\right)4$, then we can divide both sides by $2(n-4)>0$. In this case,

$$
2^{n-1}4$, then the right side is definitely smaller than the left side; hence, the inequality (1) cannot hold.

However, it is easy to verify that if the value of $n$ is $1, 2, 3$, or 4, then (1) indeed holds.