If $n$ is the smallest natural number such that $n+2n+3n+\ldots+99n$ is a perfect square,then the number of digits of $n^2$ is

Let $\{a_{n}\}_{n=1}^{\infty}$ be a sequence such that $a_{1}=1, a_{2}=1$ and $a_{n+2}=2a_{n+1}+a_{n}$ for all $n \geq 1$. Then the value of $47 \sum_{n=1}^{\infty} \frac{a_{n}}{2^{3n}}$ is equal to $.....$

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 $2$.

The sum of the first $20$ terms of the series $5+11+19+29+41+\ldots$ is $..........$.

If $1^2 + 2^2 + 3^2 + \dots + 2009^2 = (2009)(335)(4019)$ and $(1)(2009) + 2(2008) + 3(2007) + \dots + 2009(1) = (2009)(335)(x)$,then $x$ is equal to:

The value of ${1^2} + {3^2} + {5^2} + \dots + {25^2}$ is