What is the $20^{\text{th}}$ term of the sequence defined by $a_{n} = (n-1)(2-n)(3+n)$?

Let $a_{1}, a_{2}, a_{3}, \ldots$ be an $A.P.$ If $\sum_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4$,then $4 a_{2}$ is equal to

If $\frac{1^3+2^3+3^3+\ldots \text{ upto } n \text{ terms}}{1 \cdot 3+2 \cdot 5+3 \cdot 7+\ldots \text{ upto } n \text{ terms}} = \frac{9}{5}$,then the value of $n$ is

What is the sum of the series $1^2 + 2.2^2 + 3^2 + 2.4^2 + 5^2 + 2.6^2 + \dots + 2(2m)^2$?

Let $a_n$ denote the number of all $n$-digit positive integers formed by the digits $0, 1$ or both such that no consecutive digits in them are $0$. Let $b_n$ be the number of such $n$-digit integers ending with digit $1$ and $c_n$ be the number of such $n$-digit integers ending with digit $0$.
$1.$ Which of the following is correct?
$(A)$ $a_{17} = a_{16} + a_{15}$
$(B)$ $c_{17} \neq c_{16} + c_{15}$
$(C)$ $b_{17} \neq b_{16} + c_{16}$
$(D)$ $a_{17} = c_{17} + b_{16}$
$2.$ The value of $b_6$ is
$(A)$ $7$ $(B)$ $8$ $(C)$ $9$ $(D)$ $11$
Give the answer for question $1$ and $2$.

Let $f(n) = \left[ \frac{1}{3} + \frac{3n}{100} \right]n$,where $[x]$ denotes the greatest integer less than or equal to $x$. Then $\sum_{n=1}^{56} f(n)$ is equal to