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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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