Math Problem Statement

i've added more data to the question

How are the numbers in the pattern related? Use the pattern data to form a formula for the number of balls in pattern N.

Solution

The question seems to involve a pattern and a formula related to the number of balls in each pattern NN. Let's break it down step-by-step using the data in the image.


Observations:

  1. Pattern relationships:

    • Pattern 1: 4=1+34 = 1 + 3
    • Pattern 2: 7=1+67 = 1 + 6
    • Pattern 3: 10=1+910 = 1 + 9

    Here, the numbers added to 1 are 3,6,93, 6, 9, and so on.

  2. The numbers being added (3,6,93, 6, 9) are multiples of 3.

    • For N=1N = 1, 3×1=33 \times 1 = 3
    • For N=2N = 2, 3×2=63 \times 2 = 6
    • For N=3N = 3, 3×3=93 \times 3 = 9
  3. General formula: The total number of balls in pattern NN is: Number of balls=1+3N\text{Number of balls} = 1 + 3N


Questions in the image:

  1. In pattern 8: Using the formula 1+3N1 + 3N, substitute N=8N = 8: 1+3(8)=1+24=251 + 3(8) = 1 + 24 = 25 So, there are 25 balls in pattern 8.

  2. In pattern 15: Substitute N=15N = 15 into the formula: 1+3(15)=1+45=461 + 3(15) = 1 + 45 = 46 So, there are 46 balls in pattern 15.


Final Answers:

  • Formula: Number of balls=1+3N\text{Number of balls} = 1 + 3N
  • Pattern 8: 25 balls
  • Pattern 15: 46 balls

Would you like me to explain further or solve related questions? Here are 5 follow-up questions for deeper understanding:

  1. How does the formula 1+3N1 + 3N relate to arithmetic sequences?
  2. Can you derive the formula from first principles without observing the pattern?
  3. What happens to the total number of balls if NN doubles?
  4. How can you verify that the formula is correct for all values of NN?
  5. What if the pattern changes, and the added numbers are multiples of 4 instead of 3?

Tip:

When identifying patterns, look for consistent changes between terms—this often reveals the underlying rule or formula!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Patterns and Formulas

Formulas

Number of balls = 1 + 3N

Theorems

Arithmetic sequence rule

Suitable Grade Level

Grades 6-8