Suppose $1, m, n$ respectively represent the coefficient of $x^{10}$, the constant term and the coefficient of $x^{-10}$ in the expansion of $\left(a x^2+\frac{b}{x^3}\right)^{15}$. If $\frac{l}{m}+\frac{m}{n}=\frac{26}{11}$, then $a^2: b^2=$

For $z \in \mathbf{C}$, if $(1+z)^n=1+{ }^n C_1 z+{ }^n C_2 z^2+\ldots{ }^n C_n z^n$ and $\sum_{r=0}^{100} 100 c_r(\sin r x)=\left(2 \cos \frac{x}{2}\right)^{100} \sin k x$, then $k=$

If the 9th and 10th terms are the numerically greatest terms in the expansion of $(5 x-6 y)^n$ when $x=2 / 5$ and $y=1 / 2$, then the absolute value of the middle terms of that expansion is

$$ 1-\frac{3}{16}+\frac{1 \cdot 4}{1 \cdot 2}\left(\frac{3}{16}\right)^2-\frac{1 \cdot 4 \cdot 7}{1 \cdot 2 \cdot 3}\left(\frac{3}{16}\right)^3+\ldots $$