For the first one, you want to find polynomials $\displaystyle p(n), q(n)$ having integer coefficients such that

$\displaystyle p(n)(n^3+2n)+q(n)(n^4+3n^2+1)=1$.

For the third one, use the identity $\displaystyle f_n^2-f_{n-1}f_{n+1}=(-1)^{n+1}$.

For number 4, do not use wonderboy's advice, as you will *not* find a closed form for the $\displaystyle n$th harmonic number. Here is a hint : consider the highest power of $\displaystyle 2$ less than $\displaystyle n$, say $\displaystyle 2^m$. Then there is no other integer between $\displaystyle 1$ and $\displaystyle n$ which is divisible by $\displaystyle 2^m$. So what would you get if you put $\displaystyle 1+\frac{1}{2}+...+\frac{1}{n}$ in a fraction having lowest terms, say $\displaystyle \frac{p}{q}$? (show that $\displaystyle p$ is odd, while $\displaystyle q$ is even!)