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Answer :
To determine which of the given explicit equations matches the sequence [tex]\( 40, 10, \frac{5}{2}, \frac{5}{8}, \ldots \)[/tex], we need to find the pattern in the sequence and use it to identify the correct formula from the options.
First, let's observe the sequence:
1. The first term is [tex]\( 40 \)[/tex].
2. The second term is [tex]\( 10 \)[/tex].
3. The third term is [tex]\( \frac{5}{2} \)[/tex].
4. The fourth term is [tex]\( \frac{5}{8} \)[/tex].
By examining the ratio between consecutive terms:
- [tex]\( \frac{10}{40} = \frac{1}{4} \)[/tex]
- [tex]\( \frac{\frac{5}{2}}{10} = \frac{1}{4} \)[/tex]
- [tex]\( \frac{\frac{5}{8}}{\frac{5}{2}} = \frac{1}{4} \)[/tex]
Each term is obtained by multiplying the previous term by [tex]\( \frac{1}{4} \)[/tex]. This confirms that it is a geometric sequence with:
- The first term ([tex]\( a \)[/tex]) being [tex]\( 40 \)[/tex]
- The common ratio ([tex]\( r \)[/tex]) being [tex]\( \frac{1}{4} \)[/tex]
The general formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[ f(n) = a \cdot r^n \][/tex]
Where:
- [tex]\( a \)[/tex] is the first term
- [tex]\( r \)[/tex] is the common ratio
- [tex]\( n \)[/tex] is the term number starting from [tex]\( n = 0 \)[/tex] for the first term
Substituting the given values into the formula, we get:
[tex]\[ f(n) = 40 \cdot \left(\frac{1}{4}\right)^n \][/tex]
Let’s check which of the provided options matches this formula:
- [tex]\( f(n) = 40 - 30n \)[/tex] – This is not a geometric sequence formula.
- [tex]\( f(n) = 40 \left(\frac{1}{4}\right)^n \)[/tex] – This matches our derived formula.
- [tex]\( f(n) = 160 \left(\frac{1}{4}\right)^n \)[/tex] – The initial term is incorrect.
- [tex]\( f \left(r^2\right) = 40 \left(\frac{1}{2}\right)^n \)[/tex] – This does not match the structure of our sequence.
Thus, the correct explicit equation for the given sequence is:
[tex]\[ f(n) = 40 \left(\frac{1}{4}\right)^n \][/tex]
This corresponds to the option:
[tex]\[ f(n) = 40 \left(\frac{1}{4}\right)^n \][/tex]
First, let's observe the sequence:
1. The first term is [tex]\( 40 \)[/tex].
2. The second term is [tex]\( 10 \)[/tex].
3. The third term is [tex]\( \frac{5}{2} \)[/tex].
4. The fourth term is [tex]\( \frac{5}{8} \)[/tex].
By examining the ratio between consecutive terms:
- [tex]\( \frac{10}{40} = \frac{1}{4} \)[/tex]
- [tex]\( \frac{\frac{5}{2}}{10} = \frac{1}{4} \)[/tex]
- [tex]\( \frac{\frac{5}{8}}{\frac{5}{2}} = \frac{1}{4} \)[/tex]
Each term is obtained by multiplying the previous term by [tex]\( \frac{1}{4} \)[/tex]. This confirms that it is a geometric sequence with:
- The first term ([tex]\( a \)[/tex]) being [tex]\( 40 \)[/tex]
- The common ratio ([tex]\( r \)[/tex]) being [tex]\( \frac{1}{4} \)[/tex]
The general formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[ f(n) = a \cdot r^n \][/tex]
Where:
- [tex]\( a \)[/tex] is the first term
- [tex]\( r \)[/tex] is the common ratio
- [tex]\( n \)[/tex] is the term number starting from [tex]\( n = 0 \)[/tex] for the first term
Substituting the given values into the formula, we get:
[tex]\[ f(n) = 40 \cdot \left(\frac{1}{4}\right)^n \][/tex]
Let’s check which of the provided options matches this formula:
- [tex]\( f(n) = 40 - 30n \)[/tex] – This is not a geometric sequence formula.
- [tex]\( f(n) = 40 \left(\frac{1}{4}\right)^n \)[/tex] – This matches our derived formula.
- [tex]\( f(n) = 160 \left(\frac{1}{4}\right)^n \)[/tex] – The initial term is incorrect.
- [tex]\( f \left(r^2\right) = 40 \left(\frac{1}{2}\right)^n \)[/tex] – This does not match the structure of our sequence.
Thus, the correct explicit equation for the given sequence is:
[tex]\[ f(n) = 40 \left(\frac{1}{4}\right)^n \][/tex]
This corresponds to the option:
[tex]\[ f(n) = 40 \left(\frac{1}{4}\right)^n \][/tex]
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