Question

1. the graph of a certain geometric sequence can be described as a curve that is decreasing from left to right. which sequence would have a similar graph?

(a) \\( \\{1, 4, 9, 16, 25, \dots\\} \\)
(b) \\( \\{19, 16, 13, 10, \dots\\} \\)
(c) \\( \\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\\} \\)
(d) \\( \\{\frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, \dots\\} \\)

Explanation:

🆕 New Concept Discovered: Geometric Sequences
Exponential growth or decay patterns

Step 1: Understand the Goal

We need to find a sequence that satisfies two conditions:

1. It must be a geometric sequence (each term is found by multiplying the previous term by a constant number called the common ratio, \(r\)).
2. Its graph must be a curve that is decreasing from left to right (the values get smaller and smaller, flattening out toward zero).

Step 2: Analyze the Options

Let's look at each sequence to see if it is geometric and if it is decreasing.

• Option 1: \(\{1, 4, 9, 16, 25, \dots\}\)
• These are perfect squares (\(1^2, 2^2, 3^2, \dots\)).
• There is no constant multiplier between terms (\(\frac{4}{1} = 4\), but \(\frac{9}{4} = 2.25\)). This is not a geometric sequence.

• Option 2: \(\{19, 16, 13, 10, \dots\}\)
• Each term decreases by subtracting \(3\) (\(19 - 3 = 16\), \(16 - 3 = 13\)).
• Since we subtract a constant amount, this is an arithmetic sequence, not geometric. Its graph is a straight line, not a curve.

• Option 3: \(\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\}\)
• Each term is multiplied by \(\frac{1}{2}\) to get the next term:

$$ 1 \times \frac{1}{2} = \frac{1}{2}, \quad \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}, \quad \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} $$

• This is a geometric sequence with a common ratio of \(r = \frac{1}{2}\).
• Since \(0 < r < 1\), the values are positive and decreasing, forming a downward-sloping curve (exponential decay).

• Option 4: \(\{\frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, \dots\}\)
• Each term is multiplied by \(2\) to get the next term:

$$ \frac{1}{16} \times 2 = \frac{1}{8}, \quad \frac{1}{8} \times 2 = \frac{1}{4} $$

• This is a geometric sequence, but because \(r = 2\), the values are increasing (growing larger), not decreasing.

Answer:

The correct option is the third one:

$$ \{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\} $$