Proof by mathematical induction

Q.

Prove by using the principle of mathematical induction that $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{n(n+1)}=\frac{n}{n+1}$

Q.

Using the principle of mathematical induction, prove that

$\mathrm{1.2.3}+\mathrm{2.3.4}+\mathrm{3.4.5}+...+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4},$ for all $n\in N.$

Q.

If n ∈ N, then $7^{2n}$ + $2^{3n-3}$.$3^{n-1}$ is always divisible by

1. 45

2. None of these

3. 25

4. 35

Q.

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

1. 123

2. 133

3. None of these

4. 113

Q.

$1+3+3^2+\cdots +3^{n-1}=\frac{(3^n-1)}{2}$

Q. If m and n are any two odd positive integers with n < m, then the largest positive integer which divides all numbers of the form, $m^2-n^2$ can be

1. 4
2. 6
3. 8
4. 9

Q.

For every natural number n, n$(n^2-1)$ is divisible by

1. 4

2. 6

3. 10

4. None of these

Q.

Prove by the principle of mathematical induction that $(2n+7)<(n+3{)}^2$ for all natural numbers n.
Or
Prove by the principle of mathematical induction that n (n + 1) (2n + 1) is divisible by 6 for all $nϵN.$

Q.

What is the remainder when $7^{103}$ is divided by 50?

__

Q.

For all $n\ge 1$ the sum of series of 1+4+7+..+(3n-2), is equal to

1. $\frac{3^n-1}{2}$
2. $\frac{2n}{n+1}$
3. $\frac{n(3n-1)}{2}$
4. $\frac{n(3n+1)}{2}$

Q.

'For all natural numbers N, if P(n) is a statement about n and P(k+1) is true if P(k) is true for an arbitrary natural number k, then P(n) is always true.' State true or false.

1. False

2. True

Q. for N=2700 1find the total number of divisors 2 find the number of divisor divisible 15 but not by 4 3 find the sum of divisors 4 find the sum of even divisor

Q.

Let P(n) denote the statement that $n^2$ + n is odd. It

is seem that P(n) ⇒ P(n + 1), $P_n$ is true for all

1. None of these

2. n > 1

3. n

4. n > 2

Q.

'If P(n) is a statement about natural numbers, mathematical induction may be used to prove P(n) only if P(1) is true.' State true or false.

1. True

2. False

Q.

Prove that:

$(\text{cos x}-\text{cos y}{)}^2+(\text{sin x}-\text{sin y}{)}^2=4{\text{sin}}^2\frac{x-y}{2}$

Q. For any positive integer $n$, the expression $6^n-1$ is divisible by

1. 2
2. 3
3. 5
4. 7

Q. If h(x) = g(x) + f(x) is a continuous function in a given interval then g(x) and f(x) individually will also be continuous in the same interval.

1. True

2. False

Q. 22. 32n+2 - 8n 9 is divisible by 8.

Q.

Using principle of mathematical induction, prove that ${41}^n-{14}^n$ is a multiple of 27.

Or
Prove by the principle of mathematical induction $n<2^n$ for all $nϵN$.

Q.

what is the proof of the theorem:

$\left[x\right]+[x+1/n]+[x+2/n]+\dots \dots .+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=\left[nx\right]$

Q.

${}^n{C}_0-{\frac{1}{2}}^n{C}_1+{\frac{1}{3}}^n{C}_2-...........+(-1{)}^n\frac{{}^n{C}_n}{n+1}$ =

1. 1/n

2. 

3. 

4. n

Q.

Prove by the principle of mathematical induction that $\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is a natural number for all n $ϵ$ N.

Q.

Write the converse of the statement 'x is an even number implies that x is divisible by 4'

Q.

If $(1+x+2x^2{)}^{20}=a_0+a_1+a_2{x}^2+...+a_{40}{x}^{40},then{a}_1+a_3+a_5+...+a_{37}equals$ :

1. 2¹⁹(2²⁰-21)

2. 2²⁰(2¹⁹-19)

3. 2¹⁹(2²⁰+21)

4. None of these

Q.

$7^n-3^n$ is always divisible by

1. 6

2. 10

3. 8

4. 4

Q.

$P\left(n\right):a^{2n}-b^{2n}\text{is divisible by}a+b,\forall nϵN$

To prove P(n) using mathematical induction, the base case is

1. $a^1-b^1\text{is divisible by}a+b$

2. $a^2-b^2=(a+b)(a-b)\text{is divisible by}a+b$

3. $a^{2k}-b^{2k}\text{is divisible by}a+b$

4. $a^k-b^k\text{is divisible by}a+b$

Q. $\frac{d^{2}x}{d{y}^2}equals$

1. 
2. 
   
   
3. 
4. 

Q.

Find the least value of p + q if ( $p^2$ - 2p) $x^2$ + ( $q^2$ + q - 2) x = 0 is an identity.

1. 2

2. -2

3. 1

4. -1

Q. 7 + 77 + 777 + ... + 777 $\underset{n-\mathrm{digits}}{...........}7=\frac{7}{81}({10}^{n+1}-9n-10)$

Q.

$\forall nϵN,P\left(n\right):{4.7}^n+20$ is divisible by 24. State true or false.

1. True

2. False