NCERT Solutions class-11 Maths Exercise 4.1

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Exercise 4.1

Prove the following by using the principle of mathematical induction for all N:

1.

Ans. Let

For

1 = 1

is true.

Now, let be true for

……….(i)

For [Using eq. (i)]

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


2.

Ans. Let

For

1 = 1

is true.

Now, let be true for

……….(i)

For

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


3.

Ans. Let

For

1 = 1

is true.

Now, let be true for

……….(i)

For

[Using (i)]

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


4.

Ans. Let

For

6 = 6

is true.

Now, let be true for

………(i)

For

[Using eq. (i)]

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


5.

Ans. Let

For

3 = 3

is true.

Now, let be true for

For

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


6.

Ans. Let

For

2 = 2

is true.

Now, let be true for

………(i)

For

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


7.

Ans. Let

For

3 = 3

is true.

Now, let be true for

For

=

=

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


8.

Ans. Let

For

2 = 2

is true.

Now, let be true for

For

=

= =

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


9.

Ans. Let

For

is true.

Now, let be true for

For

= =

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


10.

Ans. Let

For

is true.

Now, let be true for

For

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


11.

Ans. Let

For

is true.

Now, let be true for

……….(i)

For

R.H.S. =

And L.H.S. = [Using eq. (i)]

=

=

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


12.

Ans. Let

For

is true.

Now, let be true for

……….(i)

For

R.H.S. =

L.H.S. = [Using eq. (i)]

L.H.S. =

=

=

=

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


13.

Ans. Let

For

is true.

Now, let be true for

……….(i)

For

R.H.S. =

L.H.S. = [Using eq. (i)]

L.H.S. =

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


14.

Ans. Let

For

is true.

Now, let be true for

For R.H.S. =

L.H.S. = [Using eq. (i)]

L.H.S. = =

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


15.

Ans. Let

For

is true.

Now, let be true for

……….(i)

For R.H.S. =

L.H.S. = [Using eq. (i)]

=

=

=

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


16.

Ans. Let

For

is true.

Now, let be true for

……….(i)

For R.H.S. =

L.H.S. =

L.H.S. =

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


17.

Ans. Let

For

is true.

Now, let be true for

……….(i)

For R.H.S. =

L.H.S. =

L.H.S. =

=

=

=

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


18.

Ans. Let

For

is true.

Now, let be true for

……….(i)

For ,

Now, adding on both sides of eq. (i), we have

8 < 9

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


19. is a multiple of 3.

Ans. Let is a multiple of 3.

For 1 (1 + 1) (1 + 5) is a multiple of 3 = 12 is a multiple of 3

P (1) is true.

Let be true for , is a multiple of 3.

….(i)

For , is a multiple of 3

Now,

=

= [Using (i)]

=

=

= is a multiple of 3

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


20. is divisible by 11.

Ans. Let is divisible by 11.

For is divisible by 11

= 11 is divisible by 11

P (1) is true.

Let be true for , is divisible by 11 =

……….(i)

For is divisible by 11

is divisible by 11

Now,

=

=

is divisible by 11

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


21. is divisible by

Ans. Let is divisible by

For is divisible by = is divisible by

P (1) is true.

Let be true for , is divisible by =

……….(i)

For is divisible by

Now,

=

=

= [From eq. (i)]

=

is divisible by

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


22. is divisible by 8.

Ans. Let is divisible by 8.

For is divisible by 8 = 64 is divisible by 8

P (1) is true.

Let be true for , is divisible by 8 =

……….(i)

For is divisible by 8

is divisible by 8

Now,

= [From eq. (i)]

=

= =

is divisible by 8

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


23. is a multiple of 27.

Ans. Let is a multiple of 27.

For is a multiple of 27 = 27 is a multiple of 27

P (1) is true.

Let be true for , is a multiple of 27 = …..(i)

For is a multiple of 27

Now,

=

=

= [From eq. (i)]

=

is a multiple of 27

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


24.

Ans. Let

For

9 < 16

P (1) is true.

Let be true for

……….(i)

For

=

Now, adding 2 on both sides in eq. (i),

Also

is true.

Therefore, is true.

is true.

Hence by Principle of Mathematical Induction, is true for all N.


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