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Find the recursive formula for the following sequence:

[tex]\[ 300, 150, 75, 37.5, \ldots \][/tex]

A. [tex]\( f(n) = f(n-1) + \frac{1}{2} \)[/tex]

B. [tex]\( f(n) = f(n-1) \cdot \frac{1}{2} \)[/tex]

C. [tex]\( f(n) = f(n-1) \cdot 2 \)[/tex]

D. [tex]\( f(n) = f(n-1) + 2^n \)[/tex]

E. [tex]\( f(n) = 500 \cdot 2^n \)[/tex]

Answer :

To find the recursive formula for the sequence [tex]$300, 150, 75, 37.5, \ldots$[/tex], let's follow these steps:

1. Identify the Pattern:
Look at how the sequence is changing from one term to the next.
- From [tex]$300 to 150$[/tex], the sequence is divided by 2.
- From [tex]$150 to 75$[/tex], again it is divided by 2.
- From [tex]$75 to 37.5$[/tex], it is once more divided by 2.

2. Recognize the Type of Sequence:
Since each term is obtained by multiplying the previous term by the same number (0.5), this suggests that the sequence is geometric.

3. Determine the Common Ratio:
The common ratio can be calculated as the ratio of any term to the one before it. For example:
- Common ratio between the first two terms: [tex]\(150 \div 300 = 0.5\)[/tex]
- Common ratio between the second and third terms: [tex]\(75 \div 150 = 0.5\)[/tex]
- Common ratio between the third and fourth terms: [tex]\(37.5 \div 75 = 0.5\)[/tex]

Since this ratio is consistent and equals 0.5, it confirms the sequence is geometric with a common ratio of 0.5.

4. Write the Recursive Formula:
For a geometric sequence, the recursive formula is usually given by [tex]\( f(n) = f(n-1) \times \text{common ratio} \)[/tex].

Thus, the recursive formula for this sequence is:
[tex]\[
f(n) = f(n-1) \times 0.5
\][/tex]

Therefore, among the given options, the correct recursive formula is:
[tex]\[
f(n) = f(n-1) \cdot \frac{1}{2}
\][/tex]

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