Math Problem Statement

A table is made using the following two patterns. Pattern [x]: Starting number: [12], Rule: Add by [8] Pattern [y]: Starting number: [60], Rule: Divide by [2] Complete the table for the given patterns. [x] [y] [12] [60]

Solution

Let's complete the table step by step using the given rules for each pattern:

For Pattern xx:

  • Starting number: 1212
  • Rule: Add by 88

So, each next value in the xx column is calculated by adding 88 to the previous value. Let's calculate the next few terms:

  • 12+8=2012 + 8 = 20
  • 20+8=2820 + 8 = 28
  • 28+8=3628 + 8 = 36
  • 36+8=4436 + 8 = 44

For Pattern yy:

  • Starting number: 6060
  • Rule: Divide by 22

So, each next value in the yy column is calculated by dividing the previous value by 22. Let's calculate the next few terms:

  • 60÷2=3060 \div 2 = 30
  • 30÷2=1530 \div 2 = 15
  • 15÷2=7.515 \div 2 = 7.5
  • 7.5÷2=3.757.5 \div 2 = 3.75

Completed Table:

\hline \text{Pattern } x & \text{Pattern } y \\ \hline 12 & 60 \\ 20 & 30 \\ 28 & 15 \\ 36 & 7.5 \\ 44 & 3.75 \\ \hline \end{array}$$ --- Would you like further details or clarification on any step? Here are 5 related questions for deeper understanding: 1. What happens if the rule for pattern $$x$$ changes to subtracting instead of adding? 2. How would the values in pattern $$y$$ change if the rule were to divide by 3 instead of 2? 3. How can we represent the relationship between the numbers in pattern $$x$$ using an equation? 4. What if we started with a different starting number for pattern $$y$$? How would that affect the results? 5. How can you generalize the rules for any number in a pattern like these? **Tip**: When solving problems with patterns, always check if the rule is applied consistently to each step. It can be helpful to write down the steps clearly to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Number Patterns
Arithmetic Sequences
Geometric Sequences

Formulas

Arithmetic Sequence Formula: a_n = a_1 + (n - 1) * d
Geometric Sequence Formula: a_n = a_1 * r^(n - 1)

Theorems

Properties of Arithmetic and Geometric Sequences

Suitable Grade Level

Grades 5-7