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RD Chapter 12- Mathematical Induction Ex-12.1 Interview Questions Answers

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Question 1 : If P (n) is the statement “n (n + 1) is even”, then what is P (3)?

Given:

P (n) = n (n + 1) is even.

So,

P (3) = 3 (3 + 1)

= 3 (4)

= 12

Hence, P (3) = 12, P (3) is also even.

Question 2 :

If P (n) is the statement “n3 +n is divisible by 3”, prove that P (3) is true but P (4) is not true.

Given:

P (n) = n3 + n is divisible by 3

We have P (n) = n3 + n

So,

P (3) = 33 + 3

= 27 + 3

= 30

P (3) = 30, So it is divisible by 3

Now, let’s check with P (4)

P (4) = 43 + 4

= 64 + 4

= 68

P (4) = 68, so it is not divisible by 3

Hence, P (3) is true and P (4) is not true.

Question 3 :

If P (n) is the statement “2n ≥3n”, and if P (r) is true, prove that P (r + 1) is true.

Given:

P (n) = “2n ≥ 3n” and p(r) is true.

We have, P (n) = 2n ≥ 3n

Since, P (r) is true

So,

2r≥ 3r

Now, let’s multiply both sides by 2

2×2r≥ 3r×2

2r + 1≥ 6r

2r + 1≥ 3r + 3r [since 3r>3 = 3r + 3r≥3 +3r]

2r + 1≥ 3(r + 1)

Hence, P (r + 1) is true.

Question 4 : If P (n) is the statement “n2 + n”is even”, and if P (r) is true, then P (r + 1) is true

Given:

P (n) = n2 + n is even and P (r) is true,then r2 + r is even

Let us consider r2 + r = 2k … (i)

Now, (r + 1)2 + (r + 1)

r2 + 1 + 2r + r + 1

(r2 + r) + 2r + 2

2k + 2r + 2 [from equation (i)]

2(k + r + 1)

(r + 1)2 + (r + 1) is Even.

Hence, P (r + 1) is true.

Question 5 : Given an example of a statement P (n) such that it is true for all n ϵ N.

Let us consider

P (n) = 1 + 2 + 3 + – – – – – + n = n(n+1)/2

So,

P (n) is true for all natural numbers.

Hence, P (n) is true for all n N.

Question 6 : If P (n) is the statement “n2 – n +41 is prime”, prove that P (1), P (2) and P (3) are true. Prove also that P(41) is not true.

Given:

P(n) = n2 – n + 41 is prime.

P(n) = n2 – n + 41

P (1) = 1 – 1 + 41

= 41

P (1) is Prime.

Similarly,

P(2) = 22 – 2 + 41

= 4 – 2 + 41

= 43

P (2) is prime.

Similarly,

P (3) = 32 – 3 + 41

= 9 – 3 + 41

= 47

P (3) is prime

Now,

P (41) = (41)2 – 41 + 41

= 1681

P (41) is not prime

Hence, P (1), P(2), P (3) are true but P (41) is not true.

krishan