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JEE Advanced

Explore popular questions from Principle of Mathematical Induction for JEE Advanced. This collection covers Principle of Mathematical Induction previous year JEE Advanced questions hand picked by experienced teachers.

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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

A

C

D

Explanation

Given,


From option (b),
Let


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Q 2. is divisible by

A

64

B

36

49

D

25

Explanation

Let


and are divisible by 49.
Let



Hence, by mathematical induction is divisible by 49.

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Q 3. Let is an even integer. If is assumed true is true. Therefore, is true

A

For

B

For all

C

For

None of these

Explanation

Given,
At which is not an even integer.
is not true (Principle of Induction is not applicable).
Also, is always an odd integer.

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Q 4. If is

True for all

B

True for

C

True for no

D

None of these

Explanation

Given,
(true)
Let


So, it holds for all .

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Q 5. The smallest positive integer for which holds, is

A

1

2

C

3

D

4

Explanation

Given,
At ,

At

which is true.

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Q 6. For all is divisible by

3

B

8

C

9

D

11

Explanation

For we find that takes values and which are divisible by 3

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Q 7. is divisible by for

A

B

All

D

None of these

Explanation

Let
is true.
Let is true.

Now,
(True)
So, holds for all .

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Q 8. to terms

A

B

D

None of these

Explanation

Clearly, gives the sum of the series for etc.

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Q 9. is divisible by

A

7

B

5

9

D

17

Explanation

For
which is divisible by 9.
By induction, the result is divisible by 9.

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Q 10. If , then is divisible by

A

113

B

123

133

D

None of these

Explanation

On putting in , we get

Which is divisible by 133

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Q 11. If , then is true

A

For

For

C

For

D

For all

Explanation

Given that,
Now, is true
Let

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Q 12. If then is divisible by

6

B

16

C

36

D

24

Explanation

We have,
which is product of three consecutive natural numbers and hence divisible by 6

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Q 13. For each , is divisible by

8

B

16

C

32

D

None of these

Explanation

Let
At which is divisible by 8.
is true.
Let is true, then



is divisible by 8, .

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Q 14. Let . . Then, which of the following is true?

A

is correct

C

D

Principle of mathematical induction can be used to prove the formula

Explanation

.
Put in both sides, we get
LHS and RHS

Put in both sides in the place of , we get
.

Let
Then, .


If is true, then is also true.
Hence,

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Q 15. For all is divisible by

A

25

B

26

C

1234

2304

Explanation

We have,


divisible by i.e., 2304

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Q 16. For is

A

C

D

Explanation

Let
For
For
Hence, by mathematical induction for , the proposition is true.