To determine the correct recursive formula for the sequence \(2, 7, 12, 17, 22, 27\), we start by identifying the pattern of the sequence.
- Calculate the differences between consecutive terms:
- \(7 - 2 = 5\)
- \(12 - 7 = 5\)
- \(17 - 12 = 5\)
- \(22 - 17 = 5\)
- \(27 - 22 = 5\)
The difference between each consecutive term is consistently \(5\).
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Identify the recursive formula: To express the \(n\)-th term of the sequence (\(a_n\)), we can define:
- The first term \(a_1 = 2\).
- Each subsequent term is defined by adding \(5\) to the previous term: \(a_n = a_{n-1} + 5\).
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Match with the given options: Let's examine the provided options:
- a1 = 2; an = an −1 + 7: Incorrect, the difference is 5, not 7.
- a1 = 2; an = an −1 + 5: This is correct.
- a1 = 5; an = an −1 + 2: Incorrect, first term does not match and the difference is wrong.
- a1 = 7; an = an−1 + 2: Incorrect, first term does not match and the difference is wrong.
Thus, the correct recursive formula for the sequence is \( \text{a1} = 2; \ \text{an} = \text{an−1} + 5\).
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