Math Problem Statement

How many dots are added each time in the sequence of patterns made from dots?

Solution

The problem you’ve uploaded shows a sequence of dot patterns where each pattern builds upon the previous one. You are asked to find out how many dots are added each time as the sequence progresses.

From the image:

  • Pattern number 1 has 5 dots.
  • Pattern number 3 has 13 dots.
  • Pattern number 5 has 21 dots.

The number of dots in each pattern increases according to a specific rule. The differences between consecutive terms can help us identify this pattern:

  1. From Pattern 1 (5 dots) to Pattern 3 (13 dots), the number of added dots is: 135=8 dots13 - 5 = 8 \text{ dots}

  2. From Pattern 3 (13 dots) to Pattern 5 (21 dots), the number of added dots is: 2113=8 dots21 - 13 = 8 \text{ dots}

So, it appears that 8 dots are added between each pattern in this sequence.

If you'd like, I can help explore further details about how this pattern is constructed, or if you have any questions, feel free to ask.

Would you like to explore more details, or do you have any questions?

Here are 5 related questions to think about:

  1. What would be the number of dots in Pattern 7?
  2. Can you derive a general formula for the number of dots in Pattern nn?
  3. How many dots will there be in Pattern 10?
  4. Is the increase in the number of dots consistent between every pattern?
  5. What other shapes could be formed if you continue this sequence?

Tip: When working with sequences, looking for differences between terms often helps reveal the underlying pattern!

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Math Problem Analysis

Mathematical Concepts

Sequences
Arithmetic Progression
Pattern Recognition

Formulas

Difference between terms in arithmetic sequence: a_n = a_1 + (n-1)d

Theorems

Arithmetic Progression

Suitable Grade Level

Grades 5-7