$\text{Prove that each integer}$ $n\ge3$ can be written as a sum of some consecutive natural numbers only and only if it isn't a power of 2

Find all prime numbers $ a,b,c$ that fulfill the equality: $ (a-2)!+2b!=22c-1$

Find all integers n for which number $ \log_{2n-1}(n^2+2)$ is rational.

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

For every natural number $n\geq{2}$ consider the following affirmation $P_n$: "Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots." Are $P_4$ and $P_5$ both true? Justify your answer.

Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where $n\in{N^*}$. Find the remainder of the division of polynomial $P(x^{2n+1})$ by $P(x)$.

Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]