Mathematical induction practice · Mixed difficulty

A student was asked to prove a statement $P(n)$ by induction. He proved that $P(k+1)$ is true whenever $P(k)$ is true for all $k>5 \in \mathrm{N}$ and also that $P(5)$ is true. On the basis of this he could conclude that $P(n)$ is true

If $n$ is a positive integer, then $2 \cdot 4^{2 n+1}+3^{3 n+1}$ is divisible by :

Let $P\left( n \right)$ denotes the statement that ${n^2} + n$ is odd. It is seen that $P\left( n \right) \Rightarrow P\left( {n + 1} \right),\;P\left( n \right)$ is true for all

Let $S(K)=1+3+5 \ldots+(2 K-1)=3+K^2$. Then which of the following is true

Statement $p(n): 8 n \leq 2^n-16$ is true for $n>\ldots \ldots ., n \in \mathrm{N}$.

Statement $p(n): 3^{2 n+1}+2^{n-1}, \forall \, n \in \mathrm{N}$ is divisible by

Solution of $|2 x-3| < |x+2|$ is

The least positive integer which makes $n ! < \left[\frac{n+1}{2}\right]^n$ is _____, where [ ] is the greatest integer function.

${10^n} + 3\left( {{4^{n + 2}}} \right) + 5$ is divisible by $\left( {n \in N} \right)$

Let $S\left( k \right) = 1 + 3 + 5.... + \left( {2k - 1} \right) = 3 + {k^2}$ Then, which of the following is true?

Let the number $(22)^{2022}+(2022)^{22}$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7 . Then $\left(\alpha^2+\beta^2\right)$ is equal to

If ${x^{2n - 1}} + {y^{2n - 1}}$ is divisible by $x + y$, if $n$ is

Statement I:- P(n) is a true statement $\forall n \geqslant 1;\& n \notin N$ if $P(n)$ is true for $n = t\& t + 1$ where $t \in N$ equal to set of natural numbers
Statement II:- Any statement $P(n)$ is true $\forall n \in N$ by the principle of finite mathematical
induction. Where $N =$ set of natural numbers

Let $P\left( n \right):{n^2} + n + 1$ is an even integer. If $P\left( k \right)$ is assumed true $\Rightarrow P\left( {k + 1} \right)$ is true. Therefore, $P\left( n \right)$ is true

If $n$ is a positive integer, then ${5^{2n + 2}} - 24n - 25$ isdivisible by

For all $n \geq 1$, find $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots . .+\frac{1}{n(n+1)}$

The number ${a^n} - {b^n}(a,b$ are distinct rational numbers and $n \in N$) is always divisible by

$({2^{3n}} - 1)$ will be divisible by $\left( {\forall n \in N} \right)$

Let $S_{K}=\frac{1+2+\ldots+K}{K}$ and $\sum\limits_{{j=1}}^{n} S_{j}^{2}=\frac{n}{A}\left(B n^{2}+C n+D\right)$, where $A, B, C, D \in \mathbb{N}$ and $A$ has least value. Then

The number of integral roots of the equation $|x-1|+|x+2|-|x-3|=4$ is

The sum of $\left(1^2\right)+\left(1^2+2^2\right)+\left(1^2+2^2+3^2\right)$ $+\ldots n$ brackets is equal to

$p(n): 2^{2^n}$ has _____ at unit's place for $n>1$.

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

For all $n \in N,3{n^5} + 5{n^3} + 7n$ is divisible by

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

If $n$ is a positive integer, then ${5^{2n + 2}} - 24n - 25$ isdivisible by

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

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

For all $n \in N,{7^{2n}} - 48n - 1$ is divisible by

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