Inductive Step

Q.

Let $f:\left(1,3\right)\to R$ be a function defined by $f\left(x\right)=\frac{x\left[x\right]}{\left(x^2+1\right)}$ , where$\left[x\right]$ denotes the greatest integer $\le x$. Then the range of $f$ is

1. $(\frac{2}{5},\frac{3}{5}]\cup \left(\frac{3}{4},\frac{4}{5}\right)$

2. $(\frac{2}{5},\frac{4}{5}]$

3. $\left(\frac{3}{5},\frac{4}{5}\right)$

4. $\left(\frac{2}{5},\frac{1}{2}\right)\cup (\frac{3}{5},\frac{4}{5}]$

Q. If the sum of an infinite GP $a,ar,a{r}^2,a{r}^3,...$ is $15$ and the sum of the squares of its each term is $150,$ then the sum of $a{r}^2,a{r}^4,a{r}^6,...$ is

1. $\frac{5}{2}$
2. $\frac{9}{2}$
3. $\frac{25}{2}$
4. $\frac{1}{2}$

Q. The positive integral values of n such that ${1.2}^1+{2.2}^2+{3.2}^3+{4.2}^4+{5.2}^5+.....+n{.2}^n=2^{(n+10)}+2is$

1. 313
2. 513
3. 413
4. 613

Q.

Let $S$ be a finite set containing $n$ elements. Then, the total number of commutative binary operation on $S$ is:

1. $n^3$

2. $n^2$

3. $n^{n^2}$

4. $n^n$

Q. Prove the following statement by using the principle of mathematical induction for all $n\in N$
$a+ar+a{r}^2+\cdots +a{r}^{n-1}=\frac{a(r^n-1)}{r-1}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+\cdots +\frac{1}{(1+2+3+\cdots +n)}=\frac{2n}{n+1}$

Q.

Evaluate the expression $$C(8,3)$$.

Q. Prove the following by using the principle of mathematical induction for all $n\in N$.
$1^3+2^3+3^3+\cdots +n^3={\left(\frac{n(n+1)}{2}\right)}^2$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$\frac{1}{2\cdot 5}+\frac{1}{5\cdot 8}+\frac{1}{8\cdot 11}+\cdots +\frac{1}{(3n-1)(3n+2)}=\frac{n}{(6n+4)}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$.
$1+3+3^2+\cdots +3^{n-1}=\frac{(3^n-1)}{2}$.

Q. Prove the following by using the principle of mathematical induction for all $n\in N.$
$\frac{1}{3\cdot 5}+\frac{1}{5\cdot 7}+\frac{1}{7\cdot 9}+\cdots +\frac{1}{(2n+1)(2n+3)}=\frac{n}{3(2n+3)}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$1\cdot 3+2\cdot 3^2+3\cdot 3^3+\cdots +n\cdot 3^n=\frac{(2n-1)3^{n+1}+3}{4}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$1\cdot 3+3\cdot 5+5\cdot 7+\cdots +(2n-1)(2n+1)$
$=\frac{n(4n^2+6n-1)}{3}$

Q.

Evaluate the following.

$9x^2+3x÷2+4$ for $x=5$

Q. Prove the following by using the principle of mathematical induction for all $n\in N.$
$1^2+3^2+5^2+\cdots +(2n-1{)}^2=\frac{n(2n-1)(2n+1)}{3}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N.$
$1+2+3+4+\cdots +n<\frac{1}{8}(2n+1{)}^2$.

Q. Let $A_{2\times m}$ and $B_{n\times 2}$ are two matrices such that $AB$ and $BA$ exist. If $C$ is a matrix such that the order of matrix $BAC$ is $3\times 4,$ then which of the following options is/are true

1. $m=2,n=3$
2. $m=n=3$
3. order of matrix $C$ must be $3\times 4$
4. matrix addition is not possible between the matrices $A,B$ and $C$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{2^n}=1-\frac{1}{2^n}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$(1+\frac{3}{1})(1+\frac{5}{4})(1+\frac{7}{9})\cdots (1+\frac{(2n+1)}{n^2})=(n+1{)}^2$

Q. Prove the following by using the principle of mathematical induction for all $n\in N.$
$\frac{1}{1\cdot 4}+\frac{1}{4\cdot 7}+\frac{1}{7\cdot 10}+\cdots +\frac{1}{(3n-2)(3n+1)}=\frac{n}{(3n+1)}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$\frac{1}{1\cdot 2\cdot 3}+\frac{1}{2\cdot 3\cdot 4}+\frac{1}{3\cdot 4\cdot 5}+\cdots +\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$

Q. For every positive integral value of n, $3^n>n^3$ when

1. 
2. 
3. 
4. 

Q. $1^2+2^2+\dots +n^2>\frac{n^3}{3},n\in N$

Q. If $a_1,a_2,a_3,...,a_n$ are in A.P. with $s_n$ as the sum of first 'n' terms $(S_0=0),then{\sum }_{k=0}^n{}^n{C}_k{S}_k$ is equal to

1. 
2. 
3. 
4. 

Q. For all positive integral values of n, $3^{2n}-2n+1$ is divisible by

1. 2
2. 4
3. 8
4. 12

Q. Using principle of Mathematical induction prove that:
$x^{2n}-y^{2n}$ is divisible by $x+y$, where $n\in N$

Q.

Let P(n) : $2^n<(1\times 2\times 3\times ....\times n)$. Then the smallest positive integer for which P(n) is true is

1. 1

2. 2

3. 3

4. 4

Q. Prove the following by using principle of mathematical induction for all $n\in N$ :
$1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+.......+\frac{1}{(1+2+3+\dots n)}=\frac{2n}{(n+1)}.$

Q. Let $P\left(n\right)=n^3-n$, the largest number by which $P\left(n\right)$ is divisible $\forall$ possible integral values of $n$ is

1. $2$
2. $3$
3. $5$
4. $6$