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Practice: The border Problem
Practice: The border Problem
Suppose you have an n × n square. Give two equivalent expressions that express the number of unit squares that have an edge on the boarder of the larger square.
Example Solutions:
Example Solutions:
Algebraic Equivalences:
Algebraic Equivalences:
Arithmetic Sequences:
Arithmetic Sequences:
As we can see from solution E, the sequence generated by the border problem is arithmetic sequence starting at n = 2 with initial value a2 = 4 and common difference d = 4.
Practice: The Surface Problem
Practice: The Surface Problem
Suppose you have an n × n × n cube. Give two equivalent expressions that express the number of unit cubes that have a face on the outside surface of the larger cube.
Example Solutions:
Example Solutions:
Algebraic Equivalences:
Algebraic Equivalences:
Sequences:
Sequences:
Is the sequence generated by finding the number of unit cubes with a face on the surface of the n cube an arithmetic sequence? What are some of the ways we can tell?
If we look at the first few cubes, we start with 8, then 26, then 56. The first two differences are 26 – 8 = 18 and 56 – 26 = 30.
We can also tell due to the square in the fully simplified expression that shows the closed form for the sequence is not linear.
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