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**Exercise 4.1 : ** Solutions of Questions on Page Number : **94**

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Q1 :
**
**

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

**Answer :**

Let the given statement be P(*n*), i.e.,

P(*n*): 1 + 3 + 3^{2} +
…+
3^{n}^{â€“1} =

For *n* = 1, we have

P(1): 1 =, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

1 + 3 + 3^{2} +
… +
3^{k}^{â€“1} +
3^{(}^{k}^{+1) â€“
1}

= (1 + 3 + 3^{2}
+… +
3^{k}^{â€“1}) +
3^{k}

Thus, P(*k* + 1) is true whenever P(*k*) is true.

Hence, by the principle of mathematical induction, statement
P(*n*) is true for all natural numbers i.e., *n*.

Answer needs Correction? Click Here

Q2 :
**
**

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

**Answer :**

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

P(1): 1^{3} = 1 =, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

1^{3} + 2^{3} + 3^{3} +
… +
*k*^{3} + (*k* + 1)^{3}

= (1^{3} + 2^{3} + 3^{3} +
…. +
*k*^{3}) + (*k* + 1)^{3}

Thus, P(*k* + 1) is true whenever P(*k*) is true.

Hence, by the principle of mathematical induction, statement
P(*n*) is true for all natural numbers i.e., *n*.

Answer needs Correction? Click Here

Q3 :
**
**

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

**Answer :**

Q4 :
**
**

Prove the following by using the principle of mathematical
induction for all *n*
∈ *N*: 1.2.3 + 2.3.4
+ … + *n*(*n*
+ 1) (*n* + 2) =

**Answer :**

Q5 :
**
**

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

**Answer :**

Q6 :
**
**

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

**Answer :**

Q7 :
**
**

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

**Answer :**

Q8 :
**
**

Prove the following by using the principle of mathematical
induction for all *n*
∈ *N*: 1.2 +
2.2^{2} + 3.2^{2} +
… +
*n*.2^{n} = (*n* -
1) 2^{n}^{+1} + 2

**Answer :**

Q9 :
**
**

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

**Answer :**

Q10 :
**
**

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

**Answer :**

Q11 :
**
**

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

**Answer :**

Q12 :
**
**

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

**Answer :**

Q13 :
**
**

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

**Answer :**

Q14 :
**
**

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

**Answer :**

Q15 :
**
**

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

**Answer :**

Q16 :
**
**

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

**Answer :**

Q17 :
**
**

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

**Answer :**

Q18 :
**
**

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

**Answer :**

Q19 :
**
**

Prove the following by using the principle of mathematical
induction for all *n*
∈ *N*: *n*
(*n* + 1) (*n* + 5) is a multiple of 3.

**Answer :**

Q20 :
**
**

Prove the following by using the principle of mathematical
induction for all *n*
∈ *N*:
10^{2}^{n} ^{-
1} + 1 is divisible by 11.

**Answer :**

Q21 :
**
**

Prove the following by using the principle of mathematical
induction for all *n*
∈ *N*:
*x*^{2}^{n} -
*y*^{2}^{n} is divisible by *x*
+ *y*.

**Answer :**

Q22 :
**
**

Prove the following by using the principle of mathematical
induction for all *n*
∈ *N*:
3^{2}^{n} ^{+ 2}
- 8*n* - 9 is
divisible by 8.

**Answer :**

Q23 :
**
**

Prove the following by using the principle of mathematical
induction for all *n*
∈ *N*:
41^{n} - 14^{n}
is a multiple of 27.

**Answer :**

Q24 :
**
**

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

(2*n* +7) < (*n* + 3)^{2}

**Answer :**

Maths - Maths : CBSE ** NCERT ** Exercise Solutions for Class 11th for ** Principle of Mathematical Induction ** ( Exercise 4.1 ) will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.

- Maths : Chapter 3 - Trigonometric Functions Class 11
- Maths : Chapter 5 - Complex Numbers and Quadratic Equations Class 11
- Maths : Chapter 2 - Relations and Functions Class 11
- Maths : Chapter 1 - Sets Class 11
- Maths : Chapter 9 - Sequences and Series Class 11
- Maths : Chapter 6 - Linear Inequalities Class 11
- Maths : Chapter 7 - Permutations and Combinations Class 11
- Maths : Chapter 13 - Limits and Derivatives Class 11
- Maths : Chapter 8 - Binomial Theorem Class 11