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

Answer:

Given P(n) = n(n+1)

\therefore P(3): 3 ( 3 + 1) = 12

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Question 2: If P (n) is the statement n^3 + n is divisible by 3 , prove that P(3) is true but P (4) is not true.

Answer:

Given P(n) : n^3 + n is divisible by 3

P(3) : 3^3 + 3 = 37 + 30   which is divisible by 3 \Rightarrow P(3) is true.

P(4): 4^3 + 4 = 64 + 4 = 67 which is not divisible by 3 \Rightarrow P(4) is not true.

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Question 3: If P (n) is the statement 2^n \geq 3n , and, if P(r) is true; prove that P(r + 1) is true.

Answer:

Given P(n): 2^n \geq 3n

Given P(r) is true

\Rightarrow 2^r \geq 3r

\Rightarrow 2^{r+1} \geq 6r

Since 3r \geq 3  \Rightarrow 3r+3r \geq 3+3r

\Rightarrow 2^{r+1} \geq 3(r+1)

\Rightarrow P(r+1) is true

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Question 4: If P (n) is the statement n^2+n is even, and. if P(r) is true, then P (r + 1) is true.

Answer:

Given P(n) : n^2 + n is even

Given P(r) is true \Rightarrow P(r): r^2 + r is even

\Rightarrow r^2 + r = 2\lambda where \lambda \in N

P(r+1): (r+1)^2 + ( r+1)

= r^2 + 2r + 1 + r + 1

= ( r^2 + r) + 2 ( r+1)

= 2 \lambda + 2 (r+1)

= 2 ( \lambda+r+1)

= 2 \mu   where \mu = \lambda+r+1

\Rightarrow P(r+1) is even \Rightarrow P(r+1) is true

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

Answer:

P(n): 1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2} is true for all n \in N

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

Answer:

Given P(n): n^2-n +41 is prime

P(1) : 1^2 - 1 + 41 = 41 is a prime number. Therefore P(1) is true.

P(2) : 2^2 - 2 + 41 = 43 is a prime number. Therefore P(2) is true.

P(3) : 3^2 - 3 + 41 = 47 is a prime number. Therefore P(3) is true.

P(41) : 41^2 - 41 + 41 = 41^2 is not a prime number. Therefore P(41) is not true.

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Question 7: Give an example of a statement P(n) which is true for all n \geq 4 but P(1), P(2) and P( 3) are not true. Justify your answer.

Answer:

Let P(n) : 3n < n!

P(1) : 3 \times 1 < 1! . Therefore P(1) is not true.

P(2) : 3 \times 2 < 2! . Therefore P(1) is not true.

P(3) : 3 \times 3 < 3! . Therefore P(1) is not true.

P(4) : 3 \times 4 < 4! . Therefore P(1) is true.

P(5) : 3 \times 5 < 5! . Therefore P(1) is true.

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