CBSE NCERT Solutions for Class 11 Mathematics Chapter 04

Class?XI?CBSE-Mathematics

Principle of Mathematical Induction

CBSE NCERT Solutions for Class 11 Mathematics Chapter 04

Back of Chapter Questions

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

1

+

3

+

32

+

+

3-1

=

(3 - 2

1)

Solution:

Step 1:

Considering the given statement as (),i.e.,

():

1

+

3

+

32

+

+

3-1

=

(3

- 2

1)

For = 1, we have

(1)

(31-1) 2

=

3-1 2

=

2 2

=

1.Which

is

true.

Consider, () be true for some positive integer ,i.e.,

1 + 3 + 32 + + 3-1 = (32-1)...(i) Now to prove that ( + 1)is true.

1 + 3 + 32 + + 3-1 + 3(+1)-1

= (1 + 3 + 32 + + 3-1) + 3

= (3-1) + 3

2

[Using (i)]

(3 - 1) + 2 ? 3

=

2

3(1 + 2) - 1

=

2

3 ? 3 - 1 =2

3+1 - 1 =2

Therefore, ( + 1) is true when ever () is true.

1 Practice more on Mathematical Induction



Class?XI?CBSE-Mathematics

Principle of Mathematical Induction

Therefore,( + 1)is true when ever () is true.

Thus, by the principle of mathematical induction, statement () is true for all natural numbers i.e.,.

3 OVERALL HINT: Add on both sides

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

13

+

23

+

33

+

+

3

=

( + (2

1) 2 )

Solution:

STEP 1:

Consider the given statement as (),i.e.,

():

13

+

23

+

33

+

+

3

=

( + (2

1) 2 )

For = 1, we have

(1):

13

=

1

=

(1(1+1))2

2

=

(2)2

2

=

12

=

1,which

is

true.

Let's () be true for some positive integer ,i.e.,

13 + 23 + 33 + . +3 = ((2+1))2...(i)

Now to prove that ( + 1)is true.

STEP 2:

Consider 13 + 23 + 33 + + 3 + ( + 1)3

= (13 + 23 + 33 + + 3) + ( + 1)3

= ((2+1))2 + ( + 1)3 [ Using(i) ]

=

2( + 4

1)2

+

(

+

1)3

2 Practice more on Mathematical Induction



Class?XI?CBSE-Mathematics

Principle of Mathematical Induction

2( + 1)2 + 4( + 1)3

=

4

( + 1)2{2 + 4( + 1)}

=

4

( + 1)2{2 + 4 + 4}

=

4

( + 1)2( + 2)2

=

4

( + 1)2( + 1 + 1)2

=

4

( + 1)( + 1 + 1) 2

=(

2

)

Therefore, ( + 1) is true when () is true.

Thus, by the principle of mathematical induction, statement () is true for all natural numbers i.e., .

():

13

+

23

+

33

+

+

3

=

( + (2

1) 2 )

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

1

1

1

2

1 + (1 + 2) + (1 + 2 + 3) + + (1 + 2 + 3 + ) = ( + 1)

Solution:

Step 1:

Consider the given statement as (),i.e.,

():

1

+

(1

1 +

2)

+

(1

+

1 2

+

3)

+

+

(1

+

2

1 +3

+

)

=

2 ( + 1)

For = 1,we have

(1):

1

=

2.1 1+1

=

2 2

=

1,which

is

true.

And, ()be true for some positive integer ,i.e.,

3 Practice more on Mathematical Induction



Class?XI?CBSE-Mathematics

Principle of Mathematical Induction

1

+

1 1+2

+

+

1 1+2+3

+

+

1 1+2+3++

=

2+1...(i)

Now to prove that ( + 1)is true.

STEP 2

Consider

1

1

1

1

1 + 1 + 2 + 1 + 2 + 3 + + 1 + 2 + 3 + + + 1 + 2 + 3 + + + ( + 1)

1

1

1

1

= (1 + 1 + 2 + 1 + 2 + 3 + + 1 + 2 + 3 + ) + 1 + 2 + 3 + + + ( + 1)

= 2 +

1

+1 1+2+3+++(+1)

[Using(i)]

=

2 +1

+

1 ((+1)(2+1+1))

2

2

= ( + 1) + ( + 1)( + 2)

[ 1 + 2 + 3 + + = (2+1)]

2

1

= ( + 1) ( + + 2)

2 2 + 2 + 1 = ( + 1) ( + 2 )

2 ( + 1)2 = ( + 1) [ + 2 ]

2( + 1) = ( + 2)

Therefore,( + 1)is true whenever () is true.

Thus, by the principle of mathematical induction, statement () is true for all natural numbers i.e.,.

OVER ALL HINT: TO FIND P(n); where n=1.

1

1

1

2

(): 1 + (1 + 2) + (1 + 2 + 3) + + (1 + 2 + 3 + ) = ( + 1)

4. Prove the following by using the principle of mathematical induction for all :123 + 2.3.4 +

+

(

+

1)(

+

2)

=

(+1)(+2)(+3) 4

4 Practice more on Mathematical Induction



Class?XI?CBSE-Mathematics

Principle of Mathematical Induction

Solution:

STEP1:

Consider the given statement be (),i.e.,

():

1.2.3

+

2.3.4

+

+

(

+

1)(

+

2)

=

(

+

1)( + 4

2)(

+

3)

For = 1,we have

(1):

1.2.3

=

6

=

1(1+1)(1+2)(1+3) 4

=

1.2.3.4 4

=

6,which

is

true.

Consider,() be true for some positive integer ,i.e.,

1.2.3 + 2.3.4 + + ( + 1)( + 2) = (+1)(4+2)(+3)...(i) Now to prove that ( + 1) is true.

STEP2:

Consider

1.2.3 + 2.3.4 + + ( + 1)( + 2) + ( + 1)( + 2)( + 3)

= {1.2.3 + 23.4 + + ( + 1)( + 2)} + ( + 1)( + 2)( + 3)

( + 1)( + 2)( + 3)

=

4

+ ( + 1)( + 2)( + 3)

=

(

+

1)(

+

2)(

+

3)

(4

+

1)

( + 1)( + 2)( + 3)( + 4)

=

4

( + 1)( + 1 + 1)( + 1 + 2)( + 1 + 3)

=

4

Therefore,( + 1)istruewhen()istrue.

Thus, by the principle of mathematical induction, statement () is true for all natural numbers i.e., .

():

1.2.3

+

2.3.4

+

+

(

+

1)(

+

2)

=

(

+

1)( + 4

2)(

+

3)

5. Prove the following by using the principle of mathematical induction for all :1.3 + 2.32 + 3.33 + + . 3 = (2-1)3+1+3

4

5 Practice more on Mathematical Induction



 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download