3 Marks Question

Question 13 Marks

If P(n) is the statement "n² + n is even", and if P(r) is true, then P(r + 1) is true.

Answer

P(n): n² + n is even
Given, P(r) is true
⇒ r² + r is even
$⇒ \text{r}^2 + \text{r} = 2\lambda \ ...(1)$
Now,
(r + 1)² + (r + 1)
= r² + 1 + 2r + r + 1
= (r + 1)² + 2r + 2
$=2\lambda + 2\text{r} + 2$ [Using equation (1)]
$=2(\lambda + \text{r} + 1)$
$=2\lambda$
⇒ (r + 1)² + (r + 1) is even
⇒ P(r + 1) is true

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Question 23 Marks

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.

Answer

P(n): n² - n + 41 is prime
P(1): 1 - 1 + 41 is prime
⇒ P(1): 41 is prime
$\therefore$ P(1) is true.
P(2): 2² - 2 + 41 is prime
⇒ P(2): 43 is prime
$\therefore$ P(2) is true.
P(3): 3² - 3 + 41 is prime
⇒ P(3): 47 is prime
$\therefore$ P(3) is true.
P(41): (41)² - 41 + 41 is prime
P(41): (41)² is prime
⇒ P(41) is not true.

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Question 33 Marks

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

Answer

P(n): 2ⁿ ≥ 3n
Given that P(r) is true
⇒ 2^r ≥ 3r
Multiplying both sides by 2,
2.2^r ≥ 2.3r
2^r+1 ≥ 6r
2^r+1 ≥ 3r + 3r
2^r+1 ≥ 3 + 3r, [Since 3r ≥ 3 ⇒ 3r + 3r ≥ 3 + 3r]
2^r+1 ≥ 3r(r + 1)
⇒ P(r + 1) is true.

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