Find the $7^{\text{th}}$ term in the following sequence whose $n^{\text{th}}$ term is $a_{n} = \frac{n^{2}}{2^{n}}$.

Let four distinct positive numbers $a_1, a_2, a_3, a_4$ be in a geometric progression. Let $b_1 = a_1$,$b_2 = b_1 + a_2$,$b_3 = b_2 + a_3$,and $b_4 = b_3 + a_4$.
Statement-$I$: The numbers $b_1, b_2, b_3, b_4$ are neither in an arithmetic progression nor in a geometric progression.
Statement-$II$: The numbers $b_1, b_2, b_3, b_4$ are in a harmonic progression.

Let $\langle a_n \rangle$ be a sequence such that $a_0 = 0, a_1 = \frac{1}{2}$ and $2a_{n+2} = 5a_{n+1} - 3a_n$ for $n = 0, 1, 2, 3, \ldots$. Then $\sum_{k=1}^{100} a_k$ is equal to:

Consider the sequence $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{1}=1, a_{2}=2$ and $a_{n+2}=\frac{2}{a_{n+1}}+a_{n}$ for $n=1, 2, 3, \ldots$. If $\left(\frac{a_{1}+\frac{1}{a_{2}}}{a_{3}}\right) \cdot\left(\frac{a_{2}+\frac{1}{a_{3}}}{a_{4}}\right) \cdot\left(\frac{a_{3}+\frac{1}{a_{4}}}{a_{5}}\right) \cdots\left(\frac{a_{30}+\frac{1}{a_{31}}}{a_{32}}\right)=2^{\alpha}\left({}^{61}C_{31}\right)$,then $\alpha$ is equal to.

If $a, b, c$ are in $G.P.$,and $a - b, c - a, b - c$ are in $H.P.$,then $a + 4b + c$ is equal to

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