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Consider the statement P - HSC - SSCE Mathematics Extension 2 - Question 7 - 2022 - Paper 1
Question 7
Consider the statement P. P: For all integers n ≥ 1, if n is a prime number then \( \frac{n(n + 1)}{2} \) is a prime number. Which of the following is true about t... show full transcript
Worked Solution & Example Answer:Consider the statement P - HSC - SSCE Mathematics Extension 2 - Question 7 - 2022 - Paper 1
Step 1
Evaluate the statement P
Answer
The statement P asserts that for any prime number n, the expression ( \frac{n(n + 1)}{2} ) must also yield a prime number. Let's evaluate some prime numbers:
- For n = 2: ( \frac{2(2 + 1)}{2} = \frac{6}{2} = 3 ) (which is prime)
- For n = 3: ( \frac{3(3 + 1)}{2} = \frac{12}{2} = 6 ) (which is not prime)
- For n = 5: ( \frac{5(5 + 1)}{2} = \frac{30}{2} = 15 ) (which is not prime)
From these evaluations, we see that the statement P is false as it does not hold for all prime integers.
Step 2
Evaluate the converse of the statement P
Answer
The converse of the statement P is:
"If ( \frac{n(n + 1)}{2} ) is a prime number, then n is a prime number."
To verify this, we can analyze the cases:
- If ( n = 1 ), then ( \frac{1(1 + 1)}{2} = 1 ) (which is not prime).
- If ( n = 3 ), then ( \frac{3(3 + 1)}{2} = 6 ) (which is not prime).
However, if n were prime and resulted in a prime from the formula, we would still end up producing non-prime outputs, hence the converse holds true only in specific ideal conditions it fails in many cases, so it's not guaranteed. Consequently, we determine that the converse is also false.
Step 3