Math Problem Statement
Solution
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 .)
Let's analyze the pattern:
- Figure 1 has 5 dots.
- Figure 2 has 8 dots.
- 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:
So, Figure 4 has 14 dots.
Part b: Expression for the number of dots in Figure
The general formula for an arithmetic sequence is:
Where:
- is the number of dots in Figure ,
- (the number of dots in the first figure),
- (the common difference).
So, the expression becomes:
Thus, the number of dots in Figure is given by the formula:
Do you have any questions or would you like more details?
Here are 5 follow-up questions related to this pattern and arithmetic sequences:
- What would be the number of dots in Figure 10?
- How can we find the number of dots in Figure 50 using the formula?
- If you were to plot the number of dots versus the figure number, what type of graph would you expect?
- Can you derive a recursive formula for the number of dots in Figure ?
- 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.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Patterns
Formulas
a_n = a_1 + (n - 1) * d
a_n = 3n + 2
Theorems
Arithmetic Sequence Formula
Suitable Grade Level
Grades 7-9