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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 thi... 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
To evaluate the statement P, we need to analyze the mathematical expression given. The statement claims that for any prime number ( n ), the value of ( \frac{n(n+1)}{2} ) should also be a prime number. Testing small prime numbers:
- For ( n = 2 ): ( \frac{2(2+1)}{2} = 3 ), which is prime.
- For ( n = 3 ): ( \frac{3(3+1)}{2} = 6 ), which is not prime. Thus, the statement P is false as it does not hold true for all prime n.
Step 2
Evaluate the converse of statement P
Answer
The converse of statement P states that if ( \frac{n(n+1)}{2} ) is a prime number, then ( n ) is a prime number. To evaluate this, we can find examples of numbers:
- If ( n = 3 ): ( \frac{3(3+1)}{2} = 6 ), which is not prime and does not imply that n is a prime number.
- For ( n = 5 ): ( \frac{5(5+1)}{2} = 15 ), not prime. Through further exploration, we can find various values that demonstrate this converse also does not hold for all cases. Therefore, the converse is also false.
Step 3