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

**Q1 :Prove the following by using the principle of mathematical induction for all n ∈ N:**

Answer:

Answer:

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

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 + 32 + … + 3^{k-1} + 3^{(k+1) – 1}

= (1 + 3 + 32 +… + 3^{k-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.

**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.

**Q3 :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 =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

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.

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

P(n): 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) =For n = 1, we have

P(1): 1.2.3 = 6 =, which is true.

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

1.2.3 + 2.3.4 + … + k(k + 1) (k + 2) = We shall now prove that P(k + 1) is true.

Consider

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

= {1.2.3 + 2.3.4 + … + k(k + 1) (k + 2)} + (k + 1) (k + 2) (k + 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.

**Q5 :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 = 3=, 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 + 2.32 + 3.33 + … + k3k+ (k + 1) 3

^{k+1}

= (1.3 + 2.32 + 3.33 + …+ k.3k) + (k + 1) 3

^{k+1 }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.

**Q6 :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): , 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

1.2 + 2.3 + 3.4 + … + k.(k + 1) + (k + 1).(k + 2)

= [1.2 + 2.3 + 3.4 + … + k.(k + 1)] + (k + 1).(k + 2)

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.

**Q7 :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

, 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 + 3.5 + 5.7 + … + (2k -1) (2k + 1) + {2(k + 1) -1}{2(k + 1) + 1}

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.

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

P(n): 1.2 + 2.2

^{2}+ 3.2

^{2}+ … + n.2

^{n}= (n – 1) 2

^{n+1}+ 2

For n = 1, we have

P(1): 1.2 = 2 = (1 – 1) 2

^{1+1}+ 2 = 0 + 2 = 2, which is true.

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

1.2 + 2.2

^{2}+ 3.2

^{2}+ … + k.2

^{k}= (k – 1) 2

^{k+1}+ 2 … (i)

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

Consider

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.

**Q9 :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): , 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

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.

**Q10 :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

, 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

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.

**Q11 :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

, 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

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.

**Q12 :Prove the following by using the principle of mathematical induction for all n ∈ N: **

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

For n = 1, we have

, 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

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.

**Q13 :Prove the following by using the principle of mathematical induction for all n ∈ N: **

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

For n = 1, we have

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

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

Consider

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.

**Q14 :Prove the following by using the principle of mathematical induction for all n ∈ N: **

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

For n = 1, we have

, 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

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.

**Q15 :Prove the following by using the principle of mathematical induction for all n ∈ N: **

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

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

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

Consider

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.

**Q16 :Prove the following by using the principle of mathematical induction for all n ∈ N: **

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

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

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

Consider

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.

**Q17 :Prove the following by using the principle of mathematical induction for all n ∈ N: **

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

For n = 1, we have

, 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

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.

**Q18 :Prove the following by using the principle of mathematical induction for all n ∈ N: **

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

It can be noted that P(n) is true for n = 1 since Let P(k) be true for some positive integer k, i.e.,

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

Consider

Hence,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.

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

P(n): n (n + 1) (n + 5), which is a multiple of 3.

It can be noted that P(n) is true for n = 1 since 1 (1 + 1) (1 + 5) = 12, which is a multiple of 3.

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

k (k + 1) (k + 5) is a multiple of 3.

∴k (k + 1) (k + 5) = 3m, where m ∈ N … (1)

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

Consider

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.

**Q20 :Prove the following by using the principle of mathematical induction for all n ∈ N: 10 ^{2n – 1} + 1 is divisible by 11.**

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

P(n): 10

^{2n 1}+ 1 is divisible by 11.

It can be observed that P(n) is true for n = 1 since P(1) = 10

^{2.1-1}+ 1 = 11, which is divisible by 11.

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

10

^{2k-1}+ 1 is divisible by 11.

∴10

^{2-1}+ 1 = 11m, where m ∈ N … (1)

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

Consider

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.

**Q21 :Prove the following by using the principle of mathematical induction for all n ∈ N: x ^{2n} – y^{2n} is divisible by x + y.**

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

P(n): x

^{2n}– y

^{2n}is divisible by x + y.

It can be observed that P(n) is true for n = 1.

This is so because x

^{2}

^{×1}– y

^{2}

^{×1}= x

^{2}– y

^{2}= (x + y) (x – y) is divisible by (x + y).

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

x

^{2k}– y

^{2k}is divisible by x + y.

∴x

^{2k}– y

^{2k}= m (x + y), where m ∈ N … (1)

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

Consider

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.

**Q22 :Prove the following by using the principle of mathematical induction for all n ∈ N: 3 ^{2n+2} – 8n – 9 is divisible by 8.**

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

P(n): 3

^{2n + 2}– 8n – 9 is divisible by 8.

It can be observed that P(n) is true for n = 1 since 32 ×1 + 2- 8 × 1 – 9 = 64, which is divisible by 8.

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

3

^{2k+ 2}– 8k – 9 is divisible by 8.

∴3

^{2k+ 2}– 8k – 9 = 8m; where m ∈ N … (1)

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

Consider

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.

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

P(n):41

^{n}– 14

^{n }is a multiple of 27.

It can be observed that P(n) is true for n = 1 since , which is a multiple of 27.

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

41

^{k}– 14

^{k}is a multiple of 27

∴41

^{k}– 14

^{k}= 27m, where m ∈ N … (1)

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

Consider

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.

**Q24 :Prove the following by using the principle of mathematical induction for all n ∈ N****(2n +7) < (n + 3) ^{2}**

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

P(n): (2n +7) < (n + 3)

^{2}

It can be observed that P(n) is true for n = 1 since 2.1 + 7 = 9 < (1 + 3)

^{2}= 16, which is true.

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

(2k + 7) < (k + 3)

^{2}… (1)

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

Consider

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.