Write a recursive definition for the sequence 8, 6, 4, 2, …
(1 point)
A. a1=8;an=an−1−2
B. a₁ = 2;an-1 +6
C. a₁ = 8; an an-1 + 2
D. a₁ = 2; an an-1 +8
The Correct Answer and Explanation is:
The correct recursive definition for the sequence 8, 6, 4, 2, … is:
( a_1 = 8; a_n = a_{n-1} – 2 )
Explanation:
This sequence starts at 8 and decreases by 2 at each step. To write a recursive definition, we need two parts:
- Base case: This defines the first term of the sequence. In this case, the first term is ( a_1 = 8 ).
- Recursive rule: This defines how each subsequent term relates to the previous one. In this sequence, every term is 2 less than the previous term. Hence, the recursive rule is ( a_n = a_{n-1} – 2 ).
Analyzing the Other Options:
- ( a_1 = 8; a_n = a_{n-1} – 2 ):
- This option is correct because it starts with 8 and subtracts 2 to get the next term, which matches the pattern of the sequence (8, 6, 4, 2, …).
- ( a_1 = 2; a_n = a_{n-1} + 6 ):
- This is incorrect. Starting at 2 and adding 6 each time would result in a sequence like 2, 8, 14, 20, …, which does not match the given sequence.
- ( a_1 = 8; a_n = a_{n-1} + 2 ):
- This is incorrect because adding 2 to each term would give a sequence like 8, 10, 12, …, which does not fit the decreasing pattern.
- ( a_1 = 2; a_n = a_{n-1} + 8 ):
- This is incorrect because starting at 2 and adding 8 would result in 2, 10, 18, 26, …, which again does not match the sequence.
Recursive Sequences:
A recursive sequence defines each term based on its predecessor, using a specific rule. In this case, the rule is that each term is found by subtracting 2 from the previous term. Recursive sequences are useful because they break down complex patterns into simple, repeatable steps. For this problem, the correct answer follows a straightforward approach of starting from 8 and decreasing by 2, aligning with the recursive rule ( a_n = a_{n-1} – 2 ).