https://file.aiquickdraw.com/mathbot/file/1726265014743sxeaq2oz.jpg

The task you uploaded shows three figures made up of dots (pricks) that follow a specific pattern. The questions provided are:

a. **Hur många prickar har Figur 4?**  
(How many dots are in Figure 4?)

b. **Bestäm ett uttryck för antalet prickar i Figur n.**  
(Find an expression for the number of dots in Figure \( n \).)

### Let's analyze the pattern:
1. **Figure 1** has 5 dots.
2. **Figure 2** has 8 dots.
3. **Figure 3** has 11 dots.

From this, we can observe that each figure increases by 3 dots compared to the previous one. So, the number of dots seems to follow an arithmetic sequence with a common difference of 3.

#### Part a: How many dots are in Figure 4?
The arithmetic sequence is increasing by 3 dots each time. Since Figure 3 has 11 dots, Figure 4 will have:

\[
11 + 3 = 14 \text{ dots}.
\]

So, **Figure 4** has **14 dots**.

#### Part b: Expression for the number of dots in Figure \( n \)
The general formula for an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Where:
- \( a_n \) is the number of dots in Figure \( n \),
- \( a_1 = 5 \) (the number of dots in the first figure),
- \( d = 3 \) (the common difference).

So, the expression becomes:

\[
a_n = 5 + (n - 1) \cdot 3
\]
\[
a_n = 5 + 3n - 3
\]
\[
a_n = 3n + 2
\]

Thus, the number of dots in Figure \( n \) is given by the formula:

\[
a_n = 3n + 2
\]

### Do you have any questions or would you like more details?

---

Here are 5 follow-up questions related to this pattern and arithmetic sequences:
1. What would be the number of dots in Figure 10?
2. How can we find the number of dots in Figure 50 using the formula?
3. If you were to plot the number of dots versus the figure number, what type of graph would you expect?
4. Can you derive a recursive formula for the number of dots in Figure \( n \)?
5. How would the pattern change if the common difference were 4 instead of 3?

**Tip:** Always verify patterns by checking multiple figures to ensure the rule applies consistently.

Hur många prickar har Figur 4? Bestäm ett uttryck för antalet prickar i Figur n.

Learn how to calculate the number of dots in Figure 4 and derive a general expression for Figure n in an arithmetic sequence. This problem focuses on identifying patterns in sequences and applying the arithmetic sequence formula.

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