Mathematical induction practice · Medium

A company manufactures cassettes. Its cost and revenue functions are $\mathrm{C}(\mathrm{x})=26000+30 \mathrm{x}$ and $\mathrm{R}(\mathrm{x})$ $=43 x$, respectively, where $x$ is the number of cassettes produced and sold in a week. The number of cassettes must be sold by the company to realise some profit, is

$\frac{1^4}{1 \cdot 3}+\frac{2^4}{3 \cdot 5}+\frac{3^4}{5 \cdot 7}+\ldots+\frac{n^4}{(2 n+1)(2 n-1)}$ is equal to

$\frac{3}{4}+\frac{15}{16}+\frac{63}{64}+\ldots n$ terms is equal to

If $n \in N$, then ${11^{n + 2}} + {12^{2n + 1}}$ is divisible by

$1+4+13+40+\ldots$ upto $n$ terms is equal to

If $(1)+(2+3+4)+(5+6+7+8+9)+\ldots$, then the sum of terms in the $n$th bracket is

For all $\mathrm{n} \in \mathrm{N}, 3.5^{2 \mathrm{n}+1}+2^{3 \mathrm{n}+1}$ is divisible by

For all $\mathrm{n} \in \mathrm{N}, 41^{\mathrm{n}}-14^{\mathrm{n}}$ is a multiple of___.

For natural number $n,\;{2^n}\left( {n - 1} \right)! < {n^n}$, if

If $P\left( n \right)$ is a statement such that $P\left( 3 \right)$ is true. Assuming $P\left( k \right)$ is true $\Rightarrow P\left( {k + 1} \right)$ is true for all $k \ge 3,$ then $P\left( n \right)$ is true

The solution set of the inequality $5^{x+2}>\left(\frac{1}{25}\right)^{1 / x}$

The $n\mathrm{th}$ term of the series $4 + 14 + 30 + 52 + 80 + 114 + \ldots$ is

The inequality $n! > {2^{n - 1}}$ is true for

For any $\mathrm{n} \in \mathbf{N}$, the value of the expression $\mathop {\sqrt {2 + \sqrt {2 +.....+ \sqrt 2 } } }\limits_{\mathrm{n - roots}}$ is

The solution set of $(x)^2+(x+1)^2=25$, where $(x)$ is the least integer greater than or exual to $x$, is

Solution of $|x-1| \geq|x-3|$ is

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

If ${x_1} = 3$ and ${x_{n + 1}} = \sqrt {2 + {x_n}} ,\;n \ge 1$, then $\lim\limits_{n \to \infty } {x_n}$ is equal to

$1 \cdot 2+2 \cdot 3+3 \cdot 4+\ldots$ upto $n$ terms is equal to

$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots n$ terms is equal to

For all $\mathrm{n} \in \mathrm{N}$, $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots .+\frac{1}{1+2+3+\ldots \ldots+n}$ is equal to

If $\sum\limits_{{r=1}}^k r=\frac{1}{2}\left(n^2+11 n+30\right)$, then $k=$

If $P\left( n \right):{3^n} < n!,\;n \in N$, then $P\left( n \right)$ is true

If $5^{40}$ is divided by 11 , then remainder is $\alpha$ and when $2^{2003}$ is divided by 17 , then remainder is $\beta$, then the value of $\beta-\alpha$ is

The area and perimeter of a rectangle are $A$ $P$ respectively. Then $P$ and $A$ satisfy the inequality

$1 \cdot 1 !+2 \cdot 2 !+3 \cdot 3 !+\ldots+n \cdot n !$ is equal to

${2^{3n}} - 7n - 1$ is divisible by

The set of real values of $x$ satisfying ||$x-1|-1| \mid \leq 1$ is

If $x$ satisfies the inequalities $x+7 < 2 x+3$ and $2 x+4 < 5 x+3$, then $x$ lies in the interval

For all $\mathrm{n} \in \mathrm{N}, 1.3+2.3^2+3.3^3+\ldots .+\mathrm{n} .3^{\mathrm{n}}$ is equal to