Math Problem Statement
Solution
In the image, you are asked to determine how many triangles there are in pattern number 80 from a sequence of patterns made from hexagons and triangles.
Analysis of the Problem:
- Each pattern contains hexagons and triangles.
- The image suggests that the number of triangles in each pattern increases linearly.
Looking at the patterns:
- Pattern 1 has 1 triangle.
- Pattern 2 has 3 triangles.
- Pattern 3 has 5 triangles.
- Pattern 4 has 7 triangles.
We can see that the number of triangles increases by 2 for each pattern. This forms an arithmetic sequence where the difference between consecutive terms is 2.
Formula for the nth term of an arithmetic sequence:
The nth term of an arithmetic sequence can be calculated as: Where:
- (the first term),
- (the common difference),
- (the pattern number we need to find).
Substituting the values:
Conclusion:
There are 159 triangles in pattern number 80.
Would you like more details on the arithmetic sequence or have any other questions?
Related Questions:
- What is the formula to find the number of hexagons in pattern number 80?
- How do arithmetic sequences apply to real-world problems?
- Can we derive a general formula for the total number of shapes (hexagons + triangles) in any pattern number?
- What if the number of triangles increased by a different value? How would the sequence change?
- How many triangles would be there in pattern number 100?
Tip:
Always check the pattern carefully to identify the common difference in arithmetic sequences, as this helps in forming the correct equation for the nth term.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Linear Patterns
Formulas
T_n = T_1 + (n - 1) * d
Theorems
Arithmetic Sequence Formula
Suitable Grade Level
Grades 6-8