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Principle of Mathematical Induction

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Q 1. Using mathematical induction, then numbers are defined by =1, Then, is equal to

Given,

From option (b),

Let

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Q 2. is divisible by

64

36

49

25

Let

Let

Hence, by mathematical induction

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Q 3. Let

For

For all

For

None of these

Given,

At

Also,

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Q 4. If

True for all

True for

True for no

None of these

Given,

Let

So, it holds for all

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Q 5. The smallest positive integer

1

2

3

4

Given,

At

At

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Q 6. For all

3

8

9

11

For

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Q 7.

All

None of these

Let

Let

Now,

So, holds for all

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Q 8.

None of these

Clearly,

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Q 9.

7

5

9

17

For

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Q 10. If

113

123

133

None of these

On putting

Which is divisible by 133

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Q 11. If

For

For

For

For all

Given that,

Now,

Let

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Q 12. If

6

16

36

24

We have,

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Q 13. For each

8

16

32

None of these

Let

At

Let

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Q 14. Let

Principle of mathematical induction can be used to prove the formula

Put

LHS

Put

Let

Then,

If

Hence,

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Q 15. For all

25

26

1234

2304

We have,

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Q 16. For

Let

For

For

Hence, by mathematical induction for

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