Chapter 4 Principle of Mathematical Induction Ex 4.1 Solutions
Question - 1 : - Prove the following by using the principle of mathematical induction for all n ∈ N:
Answer - 1 : -
P (k + 1) is true whenever P (k) is true.
Therefore, 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 - 2 : -
P (k + 1) is true whenever P (k) is true.
Therefore, 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 - 3 : -
P (k + 1) is true whenever P (k) is true.
Therefore, 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:
Answer - 4 : -
P (k + 1) is truewhenever P (k) is true.
Therefore, by theprinciple of mathematical induction, statement P (n) is true for all naturalnumbers i.e. n.
Prove the following by using the principle of mathematical induction for all n ∈ N:
Answer - 5 : -
P (k + 1) is true whenever P (k) is true.
Therefore, 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 - 6 : -
P (k + 1) is true whenever P (k) is true.
Therefore, 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 - 7 : -
P (k + 1) is true whenever P (k) is true.
Therefore, 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 + 2.22 + 3.22 + … + n.2n =(n – 1) 2n+1 + 2
Answer - 8 : -
We can write the givenstatement as
P (n): 1.2 +2.22 + 3.22 + … + n.2n =(n – 1) 2n+1 + 2
If n = 1 we get
P (1): 1.2 = 2 = (1 –1) 21+1 + 2 = 0 + 2 = 2
Which is true.
Consider P (k) be truefor some positive integer k
1.2 + 2.22 +3.22 + … + k.2k = (k –1) 2k + 1 + 2 … (i)
Now let us prove thatP (k + 1) is true.
Here
P (k + 1) is truewhenever P (k) is true.
Therefore, by theprinciple of mathematical induction, statement P (n) is true for all naturalnumbers i.e. n.
Question - 9 : - Prove the following by using the principle of mathematical induction for all n ∈ N:
Answer - 9 : -
We can write the givenstatement as
P (k + 1) is truewhenever P (k) is true.
Therefore, by theprinciple of mathematical induction, statement P (n) is true for all naturalnumbers i.e. n.
Question - 10 : - Prove the following by using the principle of mathematical induction for all n ∈ N:
Answer - 10 : -
P (k + 1) is truewhenever P (k) is true.
Therefore, by theprinciple of mathematical induction, statement P (n) is true for all naturalnumbers i.e. n.