A student claims that every prime greater than 3 is a term in the arithmetic sequence whose nth term is 6n + 1 or in the arithmetic sequence whose nth term is 6n – 1. Is this true? If so why?
To determine whether this claim is true, we need to analyze the two arithmetic sequences mentioned.
1. Arithmetic sequence with nth term 6n + 1: This sequence can be written as 7, 13, 19, 25, 31, 37, ...
2. Arithmetic sequence with nth term 6n - 1: This sequence can be written as 5, 11, 17, 23, 29, 35, ...
Now, let's test if every prime greater than 3 appears in either of these sequences.
Prime numbers greater than 3 can be classified into two categories:
A. Prime numbers of the form 6n + 1: e.g., 7, 13, 19, 31, 37, ...
B. Prime numbers of the form 6n - 1: e.g., 5, 11, 17, 23, 29, ...
If a prime number belongs to category A, it can be expressed as 6n + 1.
If a prime number belongs to category B, it can be expressed as 6n - 1.
Therefore, considering both sequences, every prime greater than 3 does indeed appear in either the arithmetic sequence with nth term 6n + 1 or the arithmetic sequence with nth term 6n - 1.
So, the student's claim is true, as primes greater than 3 can be represented by either of the given arithmetic sequences.