Define a sequence $\{a_n\}_{n \geq 0}$ by $a_n = \sqrt{\frac{1+a_{n-1}}{2}}$ for $n \geq 1$,with $a_0 = \cos \theta \neq \pm 1$. Then,$\lim_{n \rightarrow \infty} 4^n(1-a_n)$ equals

The minimum value of the sum of real numbers $a^{-5}, a^{-4}, 3a^{-3}, 1, a^8$ and $a^{10}$ with $a > 0$ is

Write the first three terms in each of the following sequences defined by the following: $a_{n} = \frac{n-3}{4}$

Let $x_{1}, x_{2}, x_{3}, \ldots, x_{20}$ be in a geometric progression with $x_{1} = 3$ and common ratio $r = \frac{1}{2}$. $A$ new data set is constructed by replacing each $x_{i}$ with $(x_{i} - i)^{2}$. If $\bar{x}$ is the mean of the new data,then the greatest integer less than or equal to $\bar{x}$ is $.....$

If $x = \sum_{n=0}^{\infty} (-1)^{n} \tan^{2n} \theta$ and $y = \sum_{n=0}^{\infty} \cos^{2n} \theta$ for $0 < \theta < \frac{\pi}{4}$,then:

If $a_1 = a_2 = 2$ and $a_n = a_{n-1} - 1$ for $n > 2$,then $a_5$ is