**NCERT
Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction Ex
4.1**

NCERT Solutions for Class 11 Maths Chapter 4 Principle of
Mathematical Induction Ex 4.1 are the part of NCERT Solutions for Class 11
Maths. Here you can find the NCERT Solutions for Class 11 Maths Chapter 4
Principle of Mathematical Induction Ex 4.1.

**Ex 4.1 Class 11 Maths
Question 1:**

**Ex 4.1 Class 11
Maths Question 2:**

**Solution:**

Let the given statement be
P(n), then,

P(n) = 1^{3} + 2^{3} + 3^{3} +
…………. + n^{3} = [n(n + 1)/2]^{2}.

For n = 1, we have

P(1): 1^{3} = [1(1 + 1)/2)^{2} = (2/2)^{2}

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

Let P(k) be true for some positive integer k, then,

1^{3} + 2^{3} + 3^{3} + …………. + k^{3} = [k(k + 1)/2]^{2 } …(i)

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

Consider, 1^{3} + 2^{3} + 3^{3} + ………….
+ k^{3} = [k(k + 1)/2]^{2 }

**Ex 4.1 Class 11
Maths Question 3:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 4:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 5:**

**Solution:**

**Ex 4.1 Class 11 Maths Question
6:**

**Solution:**

= k(k + 1) (k + 2)/3 + (k + 1) (k
+ 2)

= (k + 1) (k + 2) + (k/3 + 1)

= (k + 1) (k + 2) (k + 3)/3

= (k + 1) (k + 1 + 1) (k + 1 + 2)/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.

**Ex 4.1 Class 11
Maths Question 7:**

**Solution:**

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

P(n) : 1.3 + 3.5 + 5.7 + ⋅⋅⋅⋅⋅⋅⋅⋅ + (2n – 1) (2n + 1) = n(4n^{2} + 6n − 1)/3

**Ex 4.1 Class 11
Maths Question 8:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 9:**

**Solution:**

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

P(n) : ½ + ¼ + 1/8 + ⋅⋅⋅⋅⋅⋅⋅⋅ + 1/2^{n} = 1 – 1/2^{n}

**Ex 4.1 Class
11 Maths Question 10:**

**Solution:**

**Ex 4.1 Class 11 Maths Question
11:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 12:**

**Solution:**

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

P(n) : a + ar + ar^{2} + …………… + ar^{n
– 1} = a(r^{n }− 1)/(r – 1)

For n = 1, we have

P(1) : a = a(r^{1} − 1)/(r − 1) = a, which is true.

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

a + ar + ar^{2} + …………… + ar^{k – 1} = a(r^{k} − 1)/(r – 1) …………..(i)

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

Consider

{a + ar + ar^{2} + …………… + ar^{k – 1}} + ar^{(k + 1) –
1} = a(r^{k} − 1)/(r – 1) + ar^{k}
[Using equation (i)]

= [a(r^{k} − 1) + ar^{k}(r − 1)]/(r – 1)

= [ar^{k} − a + ar^{k+1} − ar^{k}]/(r – 1)

= [ar^{k+1} − a]/(r – 1)

= a(r^{k}^{+}^{1} − 1)/(r – 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.

**Ex 4.1 Class 11
Maths Question 13:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 14:**

**Solution:**

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.

**Ex 4.1 Class 11 Maths Question 15:**

Prove the following by using the principle of mathematical induction for all n ∈ N:1

^{2}+ 3

^{2}+ 5

^{2}+ … + (2n – 1)

^{2}= n(2n − 1) (2n + 1)/3

**Solution:**

**Ex 4.1 Class 11
Maths Question 16:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 17:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 18:**

**Solution:**

**Ex 4.1 Class 11
Maths Question 19:**

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

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

**Solution:**

Let the given statement be P(n), then,

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

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

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

Consider, (k + 1) {(k + 1) + 1} {(k + 1) + 5}

= (k + 1) (k + 2) {(k + 5) + 1}

= (k + 1) (k + 2) (k + 5) + (k + 1) (k + 2)

= {k (k + 1) (k + 5) + 2 (k + 1) (k + 5)} + (k + 1) (k + 2)

= 3m + (k + 1) {2 (k + 5) + (k + 2)}

= 3m + (k + 1) {2k + 10 + k + 2}

= 3m + (k + 1) (3k + 12)

= 3m + 3 (k + 1) (k + 4)

= 3 {m + (k + 1) (k + 4)}

= 3 × q, where, q = {m + (k + 1) (k + 4)} is some natural number.

Therefore, (k + 1) {(k + 1) + 1} {(k + 1) + 5} is a multiple of 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.

**Ex 4.1 Class 11
Maths Question 20:**

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

10^{2n – 1} + 1 is
divisible by 11.

**Solution:**

**Ex 4.1 Class 11
Maths Question 21:**

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

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

**Solution:**

**Ex 4.1 Class 11
Maths Question 22:**

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

3^{2n + 2} – 8n – 9
is divisible by 8.

**Solution:**

**Ex 4.1 Class 11
Maths Question 23:**

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

41^{n} – 14^{n}
is a multiple of 27.

**Solution:**

**Ex 4.1 Class 11
Maths Question 24:**

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

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

**Solution:**

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