Question

consider the geometric sequence.

(8, 4, 2, 1, \dots)

given that the sequence is represented by function (f), what are the values of (f(1)) and the common ratio?

• (f(1) = 4)

common ratio: (2)

• (f(1) = 8)

common ratio: (2)

• (f(1) = 4)

common ratio: (0.5)

• (f(1) = 8)

common ratio: (0.5)

Explanation:

🆕 New Concept Discovered: Geometric Sequences
Each term is found by multiplying the previous term by a constant.

Step 1: Find the first term \( f(1) \)

In a sequence represented by a function \( f(n) \), the term \( f(1) \) represents the first term of the sequence.

Looking at the given sequence:

$$ 8, 4, 2, 1, \dots $$

The first term is:

$$ f(1) = 8 $$

Step 2: Find the common ratio

The common ratio (\( r \)) is the multiplier used to get from one term to the next. We can find it by dividing any term by its preceding term:

$$ r = \frac{f(2)}{f(1)} = \frac{4}{8} = 0.5 $$

Answer:

• \( f(1) = 8 \)
• Common ratio: \( 0.5 \)