Final answer:
The correct explicit formula for the sequence is an = 8 - 6n and the correct recursive formula is an+1 = an - 6. Neither of these completely match the given options, indicating a potential error in the provided choices. Option D has the correct recursive formula.
Step-by-step explanation:
The student has asked for the explicit and recursive equations for the sequence 2, -4, -10, -16.... To find the explicit formula, we look for a pattern in the sequence. We notice each term is 6 less than the previous term. The first term is 2, so if we start n at 1 for the first term, the second term would be 2 - 6(1) = -4, which matches our sequence. Continuing this, the explicit formula would be an = 2 - 6(n - 1), so Option A is incorrect because its explicit formula does not match this pattern.
By simplifying our explicit formula, we can rewrite it as an = 2 - 6n + 6 or an = 8 - 6n, which is not listed in any of the options. Looking at the recursive formulas in the options, we need one that subtracts 6 from each term to match the sequence. The correct recursive formula is an+1 = an - 6, which is part of Options A and D. Since the explicit part of Option A is incorrect, we're left with Option D as the correct answer.
By confirming the sequence, we can also invalidate Options B and C because their explicit and recursive formulas do not follow the pattern of subtracting 6, but rather subtract 4, which is incorrect for this sequence.