Math Problem Statement

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

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 nn.)

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 dots.11 + 3 = 14 \text{ dots}.

So, Figure 4 has 14 dots.

Part b: Expression for the number of dots in Figure nn

The general formula for an arithmetic sequence is:

an=a1+(n1)da_n = a_1 + (n - 1) \cdot d

Where:

  • ana_n is the number of dots in Figure nn,
  • a1=5a_1 = 5 (the number of dots in the first figure),
  • d=3d = 3 (the common difference).

So, the expression becomes:

an=5+(n1)3a_n = 5 + (n - 1) \cdot 3 an=5+3n3a_n = 5 + 3n - 3 an=3n+2a_n = 3n + 2

Thus, the number of dots in Figure nn is given by the formula:

an=3n+2a_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 nn?
  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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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