Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

HARD

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If $1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + . \dots \dots + \frac{1}{n^{2}} < 2 - \frac{1}{n}$ is true for all $n > 2, n \in N$ .

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Important Questions on Mathematical Induction

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

Using mathematical induction, the numbers

a_{n}

's are defined by

a_{0} = 1, a_{n + 1} = 3 n^{2} + n + a_{n}, (n \geq 0)

, then

a_{n} =

MEDIUM

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

The least positive integral value of

λ

such that

10^{50} + (3 \times {(4)}^{52}) + λ

is divisible by

9

is

MEDIUM

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

For all

n \in ℕ, 2^{2 n + 1} + 3^{2 n + 1}

is divisible by

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

P (n) : 2 + 4 + 6 + ... + (2 n), n \in N

then

P (k) = k (k + 1) + 2

implies

P (k + 1) = (k + 1) (k + 2) + 2

is true for all

k \in N

. So, the statement

P (n) = n (n + 1) + 2

is true for

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

7^{2 n} + 16 n - 1 (n \in N)

is divisible by

HARD

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

A = [\begin{array}{l} 3 & - 4 \\ 1 & - 1 \end{array}],

then prove that

A^{n} = [\begin{array}{lr} 1 + 2 n & - 4 n \\ n & 1 - 2 n \end{array}],

where

n

is any positive integer.

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

Let

P (n)

be a statement for each natural number

n

. Assume that

P (n + 1)

is a true statement whenever

P (n)

is a true statement. Suppose

P (2018)

is true. Then which one of the following statements is true?

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

Consider the following two statements :

I. If $n$ is a composite number, then $n$ divides $(n - 1)!$

II. There are infinitely many natural numbers $n$ such that $n^{3} + 2 n^{2} + n$ divides $n!$

Then

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

Which of the following is an incorrect statement?

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

P (n) : {}^{''}2^{2 n} - 1

is divisible by

k

for all

n \in N^{''}

is true, then the value of

k

is

HARD

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

a_{n} = \sqrt{7 + \sqrt{7 + \sqrt{7 + ......}}}

having

n

radical signs, then by methods of mathematical induction which of the following is true?

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

P (n) : 2^{n} < n!

, then the smallest positive integer for which

P (n)

is true, is

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

P (n) = n (n + 1) (n + 5), n \in N

is a multiple of

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

For what natural numbers

n \in N

, the inequality

2^{n} > n + 1

is valid?

HARD

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

A = [\begin{array}{l} 1 & 0 \\ 1 & 1 \end{array}], I = [\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}],

then which of the following holds for all

n \geq 1

by the principle of induction ?

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

P (n) : 2^{n} < n!

, then the smallest positive integer for which

P (n)

is true if

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

Consider the statement: "

P (n) : n^{2} - n + 41

is prime". Then which one of the following is true?

MEDIUM

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

p

is a prime number, then

n^{p} - n

is divisible by

p

when

n

is a

HARD

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

A = [\begin{array}{l} 1 & 0 \\ 1 & 1 \end{array}]

or

[\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}]

, then which of the following holds for all

n \geq 1

, by principle of mathematical induction

EASY

Mathematics>Algebra>Mathematical Induction>The Principle of Mathematical Induction

If

P (n) : 1 + 3 + 5 + \dots + (2 n - 1) = n^{2}

is