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Write a recursive formula for the arithmetic sequence left brace lower a subscript lower n baseline right brace equals left brace 2 comma start fraction 7 over 2 end fraction comma 5 comma start fraction 13 over 2 end fraction comma ellipsis right brace.

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lower a subscript 1 baseline equals 2 semicolon lower a subscript lower n baseline equals lower a subscript n minus 1 baseline plus start fraction 3 over 2 end fraction
Image with alt text: lower a subscript 1 baseline equals 2 semicolon lower a subscript lower n baseline equals lower a subscript n minus 1 baseline plus start fraction 3 over 2 end fraction

lower a subscript 1 baseline equals 2 semicolon lower a subscript lower n baseline equals lower a subscript n minus 1 baseline plus start fraction 5 over 2 end fraction
Image with alt text: lower a subscript 1 baseline equals 2 semicolon lower a subscript lower n baseline equals lower a subscript n minus 1 baseline plus start fraction 5 over 2 end fraction

lower a subscript 1 baseline equals 2 semicolon lower a subscript lower n baseline equals start fraction 5 over 2 end fraction lower a subscript n baseline plus start fraction 3 over 2 end fraction
Image with alt text: lower a subscript 1 baseline equals 2 semicolon lower a subscript lower n baseline equals start fraction 5 over 2 end fraction lower a subscript n baseline plus start fraction 3 over 2 end fraction

lower a subscript 1 baseline equals 2 semicolon lower a subscript lower n baseline equals start fraction 3 over 2 end fraction lower a subscript n baseline plus start fraction 5 over 2 end fraction

1 answer

To find the recursive formula for the given arithmetic sequence \( a_n = { 2, \frac{7}{2}, 5, \frac{13}{2}, \ldots } \), we first need to establish the first term and the common difference.

  1. The first term \( a_1 \) is given as \( 2 \).

  2. To find the common difference, we can look at the difference between consecutive terms:

    • \( \frac{7}{2} - 2 = \frac{7}{2} - \frac{4}{2} = \frac{3}{2} \)
    • \( 5 - \frac{7}{2} = 5 - \frac{7}{2} = \frac{10}{2} - \frac{7}{2} = \frac{3}{2} \)
    • \( \frac{13}{2} - 5 = \frac{13}{2} - \frac{10}{2} = \frac{3}{2} \)

    The common difference is consistently \( \frac{3}{2} \).

Therefore, the recursive formula for the sequence can be written as: \[ a_1 = 2; \quad a_n = a_{n-1} + \frac{3}{2} \text{ for } n > 1 \]

So, the correct answer is: \[ \text{lower } a_1 = 2; \quad \text{lower } a_n = \text{lower } a_{n-1} + \frac{3}{2} \]