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Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar(s). Fractions should be reduced to lowest terms.

Consider the first three terms of the sequence below:
[tex]\[ 14,000, 12,600, 11,340, \ldots \][/tex]

Complete the recursively-defined function to describe this sequence:
[tex]\[
\begin{array}{c}
f(1) = \square, \text{ for } n \geq 2 \\
f(n) = f(n-1) \cdot \square
\end{array}
\][/tex]

The next term in the sequence is [tex]\(\square\)[/tex].



Answer :

GinnyAnswer
To describe the sequence given the first three terms [tex]\(14,000\)[/tex], [tex]\(12,600\)[/tex], and [tex]\(11,340\)[/tex], we need to complete the recursively-defined function. Let's fill in the steps:

1. Identify the first term of the sequence.
[tex]\[ f(1) = 14000 \][/tex]

2. Determine the common ratio between consecutive terms.
[tex]\[ \text{Common ratio} = \frac{12600}{14000} = 0.9 \][/tex]

3. Define the recursive formula for [tex]\(n \geq 2\)[/tex].
[tex]\[ f(n) = f(n-1) \cdot 0.9 \][/tex]

4. Calculate the next term in the sequence (the term after [tex]\(11,340\)[/tex]).
[tex]\[ \text{Next term} = 11340 \cdot 0.9 = 10206.0 \][/tex]

Summarizing these results:
[tex]\[ \begin{array}{c} f(1) = 14000, \text{ for } n \geq 2 \\ f(n) = f(n-1) \cdot 0.9 \end{array} \][/tex]

The next term in the sequence is [tex]\(10206.0\)[/tex].