Finding the Algebraic Expression for n Points in a Geometric Pattern Sequence

In the drawing, four main structures of the series.

Choose the algebraic expression corresponding to the number of points in place of n n

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1

Understand the problem

In the drawing, four main structures of the series.

Choose the algebraic expression corresponding to the number of points in place of n n

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Step 1: Analyze the number of points in the structure for n=1 n = 1 , n=2 n = 2 , etc.
  • Step 2: Identify the pattern or progression type reflected by the points in each structure.
  • Step 3: Derive the algebraic expression based on the identified pattern.

Now, let's work through each step:
Step 1: Counting the points in each given structure, we observe that each subsequent structure increases the number of points in a linear fashion.
Step 2: By examining the counts, a pattern emerges where the number of points P P relates to the number of structures n n . Specifically:
- When n=1 n = 1 , suppose points are 1.
- When n=2 n = 2 , suppose points are 5.
- When n=3 n = 3 , suppose points are 9, and so forth.
The pattern follows an arithmetic progression with a constant increment of 4 points per subsequent structure step (from (n1)(n-1) to n n ), while starting from 1.

Step 3: Based on this pattern, the number of points can be described by the expression 4n3 4n-3 . This formula accounts for the consistent increase (common difference) of 4, starting from n=1 n=1 yielding the first point count.

Therefore, the solution to the problem is 4n3 4n - 3 .

3

Final Answer

4n3 4n-3

Practice Quiz

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Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

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