Proof by mathematical induction

Q.

If n is a natural number then ${\left(\frac{n+1}{2}\right)}^n$ ≥ n ! is true

when

1. n > 1

2. n > 2

3. 1

4. 

Q.

If r is a fixed positive integer, prove by induction that (r+1)(r+2)(r+3)....(r+n) is divisible by n!

1. I want to see the solution

2. Take me to next question

Q.

Prove using mathematical induction that for all $n\ge 1$

1+4+7+..+(3n-2)=$\frac{n(3n-1)}{2}$

1. Take me to next question

2. I want to see the solution

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. n

2. None of these

3. n > 1

4. n > 2

Q.

Verify that for all $n\ge 1$ , the sum of the squares of the first 2n positive integers is given by the formula $1^2+2^2+3^2+.....(2n{)}^2=\frac{n(2n+1)(4n+1)}{3}$

1. Take me to next question

2. I want to see the solution

Q.

Use the Principle of Mathematical Induction, check if, for any positive integer $n$, $6^n-1$ is divisible by 5. True or False?

1. I want to see the solution

2. Take me to next question

Q.

For natural number n, $2^n$(n-1) ! < $n^n$, if

1. n < 2

2. n > 2

3. Never

4. 2

Q.

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

1. n > 2

2. 3

3. 4

4. n < 4

Q.

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

1. 35

2. 45

3. None of these

4. 25

Q.

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

1. 4

2. 6

3. 10

4. None of these