https://r2.erweima.ai/mathbot/image/1724322799542grrne6c3.jpg

The problem defines a sequence and asks us to find a specific term. Here are the details:

### Problem Details:
- Let \( a_n \) be the nth term of a sequence.
- The sequence has the following initial conditions:
  - \( a_1 = 12 \)
  - \( a_2 = 15 \)
  - The recurrence relation is given by:
    \[
    a_{n+2} = a_{n+1} + a_n
    \]
    for any positive integer \( n \).

We need to find \( a_5 \).

### Solution:
To find \( a_5 \), let's use the recurrence relation step-by-step.

1. **Calculate \( a_3 \)**:
   \[
   a_3 = a_2 + a_1 = 15 + 12 = 27
   \]

2. **Calculate \( a_4 \)**:
   \[
   a_4 = a_3 + a_2 = 27 + 15 = 42
   \]

3. **Calculate \( a_5 \)**:
   \[
   a_5 = a_4 + a_3 = 42 + 27 = 69
   \]

None of the options match 69. Let's check the image or confirm the problem instructions or process.

### Revised Consideration:
In this case, let's assume \( a_{n+1} = a_1 \), \( a_2 \), \( a_3 \):

We see the problem generally if the given match points continue recalculating along with the option settings. Retry otherwise, and double-check your steps from the provided image for clear alignment to this course.

### Questions:
1. What are the first 10 terms of this sequence?
2. How would the sequence change if the initial conditions were altered?
3. What is the characteristic equation of this recurrence relation?
4. Can you derive a formula for the nth term of the sequence based on the recurrence relation?
5. What happens to the sequence if we include a different recurrence relation, like \( a_{n+2} = 2a_{n+1} + a_n \)?

**Tip:** Recurrence relations are powerful tools in discrete mathematics and can model a variety of real-world processes like population growth, financial calculations, and even computer algorithms.

Explore how to solve sequence problems using recurrence relations. This example focuses on finding specific terms in a sequence defined by initial conditions and a recurrence relation. Learn step-by-step with detailed calculations and explanations. Ideal for high school students studying discrete mathematics.

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