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 + 32 +
…+
3n–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 + 32 + … + 3k–1 + 3(k+1) – 1
= (1 + 3 + 32 +… + 3k–1) + 3k
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.
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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): 13 = 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
13 + 23 + 33 + … + k3 + (k + 1)3
= (13 + 23 + 33 +
…. +
k3) + (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.
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Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
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 :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n - 1) 2n+1 + 2
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
Prove the following by using the principle of mathematical
induction for all n
∈ N: 
Answer :
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 :
Prove the following by using the principle of mathematical induction for all n ∈ N: 102n - 1 + 1 is divisible by 11.
Answer :
Prove the following by using the principle of mathematical induction for all n ∈ N: x2n - y2n is divisible by x + y.
Answer :
Prove the following by using the principle of mathematical induction for all n ∈ N: 32n + 2 - 8n - 9 is divisible by 8.
Answer :
Prove the following by using the principle of mathematical induction for all n ∈ N: 41n - 14n is a multiple of 27.
Answer :
Prove the following by using the principle of mathematical
induction for all
(2n +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.
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