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resuelve esta sucesión por favor, utiliza la formula

The problem asks how many matchsticks (phosphorus sticks) are needed to form a figure with 24 hexagons, similar to the chemical structures of benzene, naphthalene, and phenanthrene shown in the image.

Let's break it down:

1. **Benzene (Fig. 1)**: Each hexagon is made with 6 matchsticks. But when hexagons are connected (like in Fig. 2 for naphthalene or Fig. 3 for phenanthrene), some sides are shared, which reduces the total number of matchsticks needed.

2. **General Case for 1 Hexagon**: A single hexagon requires 6 matchsticks.

3. **Naphthalene (Fig. 2)**: When two hexagons are connected, one side is shared, reducing the number of matchsticks by 1. So, instead of needing \( 6 + 6 = 12 \) matchsticks, we need \( 12 - 1 = 11 \).

4. **Phenanthrene (Fig. 3)**: For three connected hexagons, each new hexagon shares one side, so the number of matchsticks is \( 6 + 5 + 5 = 16 \), not 18, because two sides are shared.

### Formula for General Case:
- For \( n \) hexagons, each additional hexagon shares one side with a previous hexagon.
- Therefore, for \( n \) hexagons, the total number of matchsticks \( M(n) \) can be calculated as:

\[
M(n) = 6 + 5(n-1) = 6 + 5n - 5 = 5n + 1
\]

Where:
- \( 6 \) is the number of matchsticks for the first hexagon.
- Each additional hexagon adds 5 matchsticks because it shares one side.

### For 24 Hexagons:
Using the formula \( M(n) = 5n + 1 \), we plug in \( n = 24 \):

\[
M(24) = 5(24) + 1 = 120 + 1 = 121
\]

So, **121 matchsticks** are needed to form a structure with 24 hexagons.

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Do you need any further details or have any questions? Here are some related questions for further exploration:

1. How would the formula change if two sides were shared between adjacent hexagons?
2. Can you extend the pattern to find the number of matchsticks for 50 hexagons?
3. How does the shape of the structure (linear, branching) affect the total number of matchsticks?
4. Can you derive the formula for a circular arrangement of hexagons?
5. How many hexagons would you need to use exactly 200 matchsticks?

**Tip:** Visualizing geometric patterns can help in developing intuition for such problems.

¿Cuántos palitos de fósforo se necesitan para formar una figura con 24 hexágonos?

Discover how to calculate the number of matchsticks required to form a structure with 24 hexagons. This problem covers geometric patterns and uses the formula M(n) = 5n + 1 to find the solution, making it ideal for students in Grades 9-12.

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