Question 1: Prove the following by using the principle of mathematical induction or all n ∈ N:
ANSWER: Let the given statement be P(n), i.e.,
P(n): 1 +3+ 3^{2} … 3^{n}^{–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}^{–1} + 3^{(k 1) – 1}
= (1 + 3+ 3^{2} … 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.
Question 2: 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.
Question 3: 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.
Question 4: 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.
Question 5: 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+ 2.3^{2} + 3.3^{3} … +k3^{k} +(k +1) 3^{k}^{ 1}
= (1.3+ 2.3^{2} + 3.3^{3} … +k.3^{k})+ (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.
Question 6: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.
Question 7: 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.
Question 8: 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.
Question 9: 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.
Question 10: 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.
Question 11: 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.
Question 12: 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.
Question 13: 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.
Question 14: 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.
Question 15: 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.
Question 16: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.
Question 17: 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.
Question 18: 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.
Question 19: 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.
Question 20: 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^{2k – 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.
Question 21: 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.
Question 22: 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 3^{2 × 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.
Question 23: 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.
Question 24: Prove the following by using the principle of mathematical induction for all (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.
75 videos238 docs91 tests

1. How do you prove a statement using the principle of mathematical induction? 
2. What is the base case in a proof by mathematical induction? 
3. Is mathematical induction the only way to prove mathematical statements? 
4. Can mathematical induction be used to prove inequalities? 
5. How can mathematical induction be applied to solve problems in combinatorics or number theory? 

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