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The problem shows a geometric pattern constructed using matchsticks, where each picture builds on the previous one. We need to determine how many matchsticks are in the 50th picture of this sequence.

### Observing the pattern:
- The first figure is a triangle, which consists of **3 matchsticks**.
- The second figure adds another triangle, but **shares one side** with the previous triangle, so it uses **5 matchsticks**.
- The third figure adds yet another triangle, sharing one side, and uses **7 matchsticks**.

Thus, it appears that each new figure in the sequence adds **2 matchsticks** (since only two sides of the new triangle require additional matchsticks).

### General Formula:
The number of matchsticks in the \(n\)-th figure can be described by the following linear formula:
\[
M(n) = 3 + 2(n - 1)
\]
Where:
- \(M(n)\) is the number of matchsticks in the \(n\)-th figure.
- \(n\) is the figure number.
- \(3\) is the number of matchsticks in the first figure.
- \(2(n - 1)\) accounts for the additional matchsticks added to each new figure.

### Applying to the 50th figure:
\[
M(50) = 3 + 2(50 - 1) = 3 + 2(49) = 3 + 98 = 101
\]

Thus, the 50th figure will have **101 matchsticks**.

Would you like further clarification or details on the steps? Here are a few related questions you could explore:

1. How can you generalize the formula to other shapes?
2. How would the pattern change if the matchsticks did not share sides?
3. What is the total number of matchsticks used in all 50 figures combined?
4. How would you modify the formula if two matchsticks were removed in every alternate step?
5. Can this problem be solved using a recursive approach?

**Tip**: When solving patterns, look for consistent changes (like "adding 2 matchsticks") to find a linear relationship.

How many matchsticks are in the 50th picture in this pattern?

Learn how to calculate the number of matchsticks in the 50th figure of a geometric pattern using a simple linear formula. This problem covers arithmetic progression concepts and is suitable for students in Grades 6-8.

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