Given the sequence -14, -6, -2, 0, 1, ..., find the recursive formula.

A. a1=−14�1=−14 and an+1=10an+6��+1=10��+6

B. a1=−14�1=−14 and an+1=2+an2��+1=2+��2

C. a1=−14�1=−14 and an+1=an+n��+1=��+�

D. a1=−14�1=−14 and an+1=nan


To determine which of the given options correctly represents the recursive formula for the sequence \(-14, -6, -2, 0, 1, \dots\), we can test each formula to see which one generates the sequence correctly.

Given sequence: \(-14, -6, -2, 0, 1, \dots\)

### Option A: \(a_1 = -14\) and \(a_{n+1} = 10a_n + 6\)
- \(a_1 = -14\)
- \(a_2 = 10(-14) + 6 = -140 + 6 = -134\) (this does not match the sequence)

This option is incorrect.

### Option B: \(a_1 = -14\) and \(a_{n+1} = \frac{2 + a_n}{2}\)
- \(a_1 = -14\)
- \(a_2 = \frac{2 + (-14)}{2} = \frac{-12}{2} = -6\)
- \(a_3 = \frac{2 + (-6)}{2} = \frac{-4}{2} = -2\)
- \(a_4 = \frac{2 + (-2)}{2} = \frac{0}{2} = 0\)
- \(a_5 = \frac{2 + 0}{2} = \frac{2}{2} = 1\)

This matches the given sequence, so this option is correct.

### Option C: \(a_1 = -14\) and \(a_{n+1} = a_n + n\)
- \(a_1 = -14\)
- \(a_2 = -14 + 1 = -13\) (this does not match the sequence)

This option is incorrect.

### Option D: \(a_1 = -14\) and \(a_{n+1} = n \times a_n\)
- \(a_1 = -14\)
- \(a_2 = 1 \times (-14) = -14\) (this does not match the sequence)

This option is incorrect.

### Conclusion:
The correct answer is **Option B**: \(a_1 = -14\) and \(a_{n+1} = \frac{2 + a_n}{2}\).

Would you like to delve into the derivation of this formula, or do you have any other questions?

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Here are some related questions to explore:

1. How can you generalize the method used to find this recursive formula for other sequences?
2. What would the explicit formula for this sequence look like?
3. How would the sequence change if the initial term \(a_1\) was different?
4. Can you identify other sequences that follow a similar recursive pattern?
5. How can you check if a sequence is arithmetic or geometric?
6. What happens to this sequence as \(n\) becomes very large?
7. Can you derive a closed-form expression for this sequence?
8. What is the importance of the initial condition in recursive sequences?

**Tip:** When dealing with recursive sequences, it's often helpful to calculate the first few terms to see if the pattern matches the given sequence.

Discover how to find the recursive formula for the sequence -14, -6, -2, 0, 1, ... through detailed analysis. Explore the correct application of recursive formula options and their implications. Suitable for advanced high school students.

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