We are tasked with finding the first four terms of a sequence defined recursively by:

• $a_1 = -14$
• $a_n = \frac{2 + a_{n-1}}{2}$ for $n \geq 2$.

We will calculate $a_2, a_3,$ and $a_4$ step by step.

Step 1: Calculate $a_2$

Using the formula $a_n = \frac{2 + a_{n-1}}{2}$:
$a_2 = \frac{2 + a_1}{2} = \frac{2 + (-14)}{2} = \frac{-12}{2} = -6.$

Step 2: Calculate $a_3$

$a_3 = \frac{2 + a_2}{2} = \frac{2 + (-6)}{2} = \frac{-4}{2} = -2.$

Step 3: Calculate $a_4$

$a_4 = \frac{2 + a_3}{2} = \frac{2 + (-2)}{2} = \frac{0}{2} = 0.$

Final Answer

The first four terms of the sequence are:
$a_1 = -14, \quad a_2 = -6, \quad a_3 = -2, \quad a_4 = 0.$

Would you like further details or an explanation of the method?

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Related Questions:

1. What is the behavior of this sequence as $n \to \infty$?
2. Can this sequence be expressed in a closed form (non-recursive formula)?
3. How can we verify that the sequence converges?
4. What is the limit of the sequence if it converges?
5. How would changing $a_1$ affect the first four terms?

Tip:

Recursive sequences often converge if the recursive formula is based on a contractive mapping (e.g., averages like this one).