Mathematical Induction

Let \(P(n): 2+2^2+2^3+\ldots+2^n=2^{n+1}, n \in \mathbf{N}\).$ Then,

$\sum_{k=1}^{2 n+1}(-1)^{k-1} \cdot k^2$ equals to

For what natural numbers $n\in\,N$, the inequality $2^{n}>n+1$is valid?

$\sum_{n=1}^5 n\left(n^2+n+1\right)=$

If $2^{\mathrm{n}}$ divides 16 ! and $2^{\mathrm{n}+1}$ does not divide 16 !, then $\mathrm{n}=$

Using mathematical induction, the numbers $a_{n}$'s are defined by$a_{0}=1,a_{n+1}=3n^{2}+n+a_{n},\left(n\geq\,0\right)$, then $a_{n}=$

If $2\left(4^{2 n+1}\right)+3^{3 n+1}$ is divisible by $k, k>1$ for all $n \in N$, then the value of $k$ is

Using mathematical induction, the numbers $a_n$ 's are defined by, $a_0=1, a_{n+1}=3 n^2+n+a_n,(n \geq 0) .$ Then, $a_n$ is equal to

The given following circuit is equivalent to

The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of objective function $3 x+5 y$, subject to the linear constraints given by this system of inequations is

If ' $n$ ' is a positive integer, then $\mathrm{n}^{3}+2 \mathrm{n}$ is divisible by

If \(P(n): 2^{n} < n !\) then the smallest positive integer for which \(P(n)\) is true, is