If $n$ is a positive integer and $f(n)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$, then $f(2023)=$

If $y=\frac{3}{4}+\frac{3 \cdot 5}{4 \cdot 8}+\frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12}+\ldots$ to $\infty$, then

The numerically greatest term in the binomial expansion of $(2 x-3 y)^5$, when $x=\frac{3}{2}$ and $y=\frac{2}{3}$ is

If $\frac{2 x^3+3 x^2+3 x+5}{\left(x^2+1\right)\left(x^2+2\right)}$ is expanded in terms of the powers of $x$, then the coefficient of $x^5$ is