Math Problem Statement

The start of a sequence of patterns made from dots and sticks is shown below. The same number of dots and sticks is added each time. Write an expression for the number of dots and sticks in the nth pattern.

Solution

It looks like you've uploaded three images related to pattern sequences. They seem to involve sequences of dots and sticks, where a certain number of items is added to each subsequent pattern. Let me help you with the questions based on these patterns.

First Image: Dots Pattern

In the first image, the number of dots increases as the pattern number increases. Here's the breakdown:

  • Pattern 1 has 3 dots.
  • Pattern 2 has 5 dots.
  • Pattern 3 has 7 dots.

This suggests the number of dots increases by 2 each time.

a) Expression for the number of dots in the nth pattern:

Since the number of dots starts at 3 and increases by 2 each time, we can express the number of dots in the nth pattern as: Number of dots in nth pattern=2n+1\text{Number of dots in nth pattern} = 2n + 1

b) Number of dots in the 13th pattern:

Substitute n=13n = 13 into the formula: 2(13)+1=26+1=272(13) + 1 = 26 + 1 = 27 So, there are 27 dots in the 13th pattern.


Second Image: Sticks Pattern

The second image involves a sequence of stick patterns, and each pattern has more sticks than the previous one. Let's break it down:

  • Pattern 1 has 3 sticks.
  • Pattern 2 has 5 sticks.
  • Pattern 3 has 7 sticks.

Again, we observe that the number of sticks increases by 2 each time.

Rule for the number of sticks in the nth pattern:

The number of sticks in the nth pattern can be expressed as: Number of sticks in nth pattern=2n+1\text{Number of sticks in nth pattern} = 2n + 1


Third Image: Dots Pattern

The third image shows another sequence of dots, but in this case:

  • Pattern 1 has 3 dots.
  • Pattern 2 has 5 dots.
  • Pattern 3 has 7 dots.

The same pattern of increasing by 2 each time is visible.

Expression for the number of dots in the nth pattern:

Similar to the first pattern, the expression for the number of dots in the nth pattern is: Number of dots in nth pattern=2n+1\text{Number of dots in nth pattern} = 2n + 1


Feel free to ask for more detailed explanations, or if you need help with other problems! Would you like to explore this topic further?

Further questions to explore:

  1. How can we apply the formula 2n+12n + 1 to other patterns like squares or triangles?
  2. What happens if the number of dots or sticks increases by a different number, say 3 or 4 each time?
  3. How do arithmetic sequences relate to these types of patterns?
  4. How do you graph patterns based on arithmetic sequences like these?
  5. How would you find the nth term in a geometric sequence, and how does it differ from an arithmetic one?

Tip:

The sequence 2n+12n + 1 is an example of an arithmetic sequence where the first term is 3, and the common difference is 2. Recognizing the pattern's structure can help simplify solving similar problems quickly!

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Math Problem Analysis

Mathematical Concepts

Patterns
Sequences
Arithmetic Sequences
Algebra

Formulas

Number of dots in nth pattern = 2n + 1
Number of sticks in nth pattern = 2n + 1

Theorems

Arithmetic sequence formula

Suitable Grade Level

Grades 6-8