Consider the statement: "P(n): n2 – n + 41 is prime. "The which one of the following is true?
P(n) = n2 + 41
P(3) = 9 – 3 + 41= 47
P(5) = 25 – 5 + 41= 61
Hence P(3) and P(5) are both prime
Let's analyze the statement: P(n): n2 – n + 41 is prime.
We need to check P(3) and P(5) by substituting n=3 and n=5 into the expression.
Step 1: Evaluate P(3)
Substitute n=3:
47 is a prime number (divisible only by 1 and 47). So, P(3) is true.
Step 2: Evaluate P(5)
Substitute n=5:
61 is a prime number (divisible only by 1 and 61). So, P(5) is also true.
Conclusion: Both P(3) and P(5) are true. Therefore, the correct option is: "Both P(3) and P(5) are true".
Note: Although this expression generates primes for many small n (like n=0 to 40), it is not prime for all n. For example, at n=41, P(41)=412 – 41 + 41 = 1681, which is not prime (41×41). But for n=3 and n=5, it indeed gives primes.
Prime Numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, ...
Quadratic Expressions: Expressions of the form ax2 + bx + c. Here, n2 – n + 41 is a quadratic in n.
Mathematical Induction: Sometimes used to prove statements for all natural numbers, but note that P(n) here is not true for all n (as seen at n=41).