Question

1. is the sequence \\( (81, 27, 9, 3, 1, \dots) \\) arithmetic or geometric?

• arithmetic because the common ratio is \\( \frac{1}{3} \\).
• geometric because the common ratio is \\( \frac{1}{3} \\).
• arithmetic because the common difference is \\( 3n \\).
• geometric because the common difference is \\( 3n \\).

Explanation:

🆕 New Concept Discovered: Arithmetic vs. Geometric Sequences
Multiplying vs. adding to find the next term

Step 1: Analyze the sequence behavior

Let's look at the given sequence:

$$ (81, 27, 9, 3, 1, \dots) $$

To find how we get from one term to the next, we check the relationship between consecutive terms.

• Arithmetic sequences change by adding or subtracting a constant value (a common difference).
• Geometric sequences change by multiplying or dividing by a constant value (a common ratio).

Step 2: Test for a common ratio

Let's divide each term by the term before it to see if there is a constant multiplier (common ratio, \( r \)):

$$ \frac{27}{81} = \frac{1}{3} $$

$$ \frac{9}{27} = \frac{1}{3} $$

$$ \frac{3}{9} = \frac{1}{3} $$

$$ \frac{1}{3} = \frac{1}{3} $$

Since we multiply each term by \(\frac{1}{3}\) to get the next term, the sequence is geometric with a common ratio of \(\frac{1}{3}\).

Answer:

Geometric because the common ratio is \(\frac{1}{3}\).