The sequence defined recursively by \( a_1 = 6 \) and \( a_n = a_{n-1} + 2 \) describes a sequence where each term is 2 more than the previous term, starting from 6.
Let's compute the first few terms:
- \( a_1 = 6 \)
- \( a_2 = a_1 + 2 = 6 + 2 = 8 \)
- \( a_3 = a_2 + 2 = 8 + 2 = 10 \)
- \( a_4 = a_3 + 2 = 10 + 2 = 12 \)
- \( a_5 = a_4 + 2 = 12 + 2 = 14 \)
- \( a_6 = a_5 + 2 = 14 + 2 = 16 \)
This gives us the sequence: \( 6, 8, 10, 12, 14, 16, \ldots \)
Now, let's analyze the provided options:
- \( a_n = 8, 10, 12, 14, 16, \ldots \) - This does not start at 6.
- \( a_n = 6, 4, 2, 0, -2, \ldots \) - This sequence decreases, so it does not match.
- \( a_n = 4 + 2n \) - If \( n = 1 \), then \( a_1 = 4 + 2(1) = 6 \), but this doesn’t match the pattern of the generated sequence from the recursion.
- \( a_n = 6 + 2n \) - For \( n = 1 \), \( a_1 = 6 + 2(1) = 8 \), but it becomes \( 6 + 2(2) = 10 \), etc. So this does not match when \( n = 1 \).
None of the fourth choices accurately represent the generated sequence.
The correct form for the sequence given by the recursive definition is \( a_n = 6 + 2(n - 1) = 2n + 4 \), which matches the recursive description.
Based on available responses, the one that is most similar considering the base term will be option 4 with \( a_n = 6 + 2n \) (considering that if you start at n=0, then the first term aligns).
However, specifically, it would be \( a_n = 6, 8, 10, 12, 14, 16, \ldots \) which is formatted incorrectly in the available choices.
If only that sequence were presented, it would be the correct response.
1 answer