Let $a > b > 0$ and $f(n) = a^{1/n} - b^{1/n}$, $J(n) = (a - b)^{1/n}$ for all $n \geq 2$. Then:

The coefficient of $x^{15}$ in the product $(1-x)(1-2x)(1-2^2x)(1-2^3x) \ldots (1-2^{15}x)$ is

Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^{P} \quad(P \in N)$ in the expansion of $\frac{1}{(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})}$ is $\alpha_{P}$,then $\alpha_{K}-\alpha_{K+1}-\alpha_{K-1}=$

$1+{ }^{n} C_{1} \cos \theta+{ }^{n} C_{2} \cos 2 \theta+\ldots+{ }^{n} C_{n} \cos n \theta$ equals

For an integer $n \geq 2$,if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2n-3}$ is $16$,then the distance of the point $P(2n-1, n^2-4n)$ from the line $x+y=8$ is:

Let $\lambda$ be the positive root of the equation $x^2-x-1=0$,and set $a_n = \frac{1}{\sqrt{5}}\left(\lambda^n - (1-\lambda)^n\right)$ for $n \in N$,where $N$ is the set of all natural numbers. Consider the sets $A = \{ n \in N : a_n \text{ is a rational number, but not an integer} \}$ and $B = \{ n \in N : a_n \text{ is an irrational number} \}$. Then: