patterns exist in nature and in everyday life. patterns occur over and over again; patterns show relationships and trends.
some examples of patterns in nature are:
example 1
draw the 4th figure in the following pattern.
example 2
complete the following number patterns:
2, 4, 6,
is this pattern increasing or decreasing? by how much?
0, 6, 12,
is this pattern increasing or decreasing? by how much?
900, 800, 700,
is this pattern increasing or decreasing? by how much?
example 3
state the pattern rules.
pattern rule:
pattern rule:

Question

Accepted Answer

### Example 1:
# Explanation:
## Step1: Analyze the pattern of figures
The first figure (1st) has 3 squares (2 in the first column, 1 in the second), the second has 6 squares (3 in the first column, 2 in the second, 1 in the third), the third has 10 squares (4 in the first column, 3 in the second, 2 in the third, 1 in the fourth). The number of squares in each column for the nth figure: column 1 has $n + 1$ squares, column 2 has $n$ squares, ..., column $n+1$ has 1 square. For the 4th figure, column 1: $4 + 1=5$ squares, column 2: $4$ squares, column 3: $3$ squares, column 4: $2$ squares, column 5: $1$ square. So we draw a figure with 5 squares in the first column, 4 in the second, 3 in the third, 2 in the fourth, 1 in the fifth (stacked vertically, each column to the right has one less square than the previous).

## Step2: Sketch the 4th figure
(Visual description: A stepped figure where the first column (leftmost) has 5 squares stacked, the second column has 4 squares (aligned with the bottom of the first column's 5), the third has 3, the fourth has 2, the fifth has 1. So it looks like a larger "staircase" than the third figure, with one more column (fifth) and each existing column having one more square than the third figure's columns.)

# Answer: The 4th figure is a stepped arrangement with 5 squares in the first column, 4 in the second, 3 in the third, 2 in the fourth, and 1 in the fifth (drawn as a vertical stack per column, right columns shorter by 1 square each from the left).

### Example 2 (First Number Pattern: 2, 4, 6, ...):
# Explanation:
## Step1: Identify the pattern
The difference between consecutive terms: $4 - 2 = 2$, $6 - 4 = 2$. So each term increases by 2.

## Step2: Find the next terms
After 6, the next term is $6 + 2 = 8$, then $8 + 2 = 10$, then $10 + 2 = 12$.

## Step3: Determine trend and difference
The pattern is increasing because each term is larger than the previous. The increase is by 2.

# Answer: 8, 10, 12; Increasing, by 2.

### Example 2 (Second Number Pattern: 0, 6, 12, ...):
# Explanation:
## Step1: Identify the pattern
Difference between terms: $6 - 0 = 6$, $12 - 6 = 6$. So each term increases by 6.

## Step2: Find the next terms
After 12, next term: $12 + 6 = 18$, then $18 + 6 = 24$, then $24 + 6 = 30$.

## Step3: Determine trend and difference
Pattern is increasing, by 6.

# Answer: 18, 24, 30; Increasing, by 6.

### Example 2 (Third Number Pattern: 900, 800, 700, ...):
# Explanation:
## Step1: Identify the pattern
Difference between terms: $800 - 900 = -100$, $700 - 800 = -100$. So each term decreases by 100.

## Step2: Find the next terms
After 700, next term: $700 - 100 = 600$, then $600 - 100 = 500$, then $500 - 100 = 400$.

## Step3: Determine trend and difference
Pattern is decreasing, by 100.

# Answer: 600, 500, 400; Decreasing, by 100.

### Example 3 (First Table: In - Out):
# Explanation:
## Step1: Analyze In and Out values
For In = 11, Out = 22: $22 - 11 = 11$; In = 23, Out = 34: $34 - 23 = 11$; In = 35, Out = 46: $46 - 35 = 11$; In = 43, Out = 54: $54 - 43 = 11$; In = 48, Out = 59: $59 - 48 = 11$.

## Step2: State the pattern rule
Output = Input + 11.

# Answer: Pattern Rule: Output is Input plus 11 (Output = Input + 11).

### Example 3 (Second Table: Input - Output):
# Explanation:
## Step1: Analyze Input and Output values
For Input = 19, Output = 12: $19 - 12 = 7$; Input = 21, Output = 14: $21 - 14 = 7$; Input = 30, Output = 23: $30 - 23 = 7$; Input = 40, Output = 33: $40 - 33 = 7$; Input = 46, Output = 39: $46 - 39 = 7$.

## Step2: State the pattern rule
Output = Input - 7.

# Answer: Pattern Rule: Output is Input minus 7 (Output = Input - 7).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Example 1:

Step1: Analyze the pattern of figures

Step2: Sketch the 4th figure

Step1: Identify the pattern

Step2: Find the next terms

Step3: Determine trend and difference

Step1: Identify the pattern

Step2: Find the next terms

Step3: Determine trend and difference

Answer:

Example 2 (First Number Pattern: 2, 4, 6, ...):