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Answer :
To determine which formula can describe the sequence [tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots\)[/tex], we need to understand the nature of the sequence. The sequence appears to be geometric, which means each term should be a constant multiple, called the common ratio, of the previous term.
Let's first convert the sequence terms into improper fractions for easier handling:
1. [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
2. [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
3. [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
4. [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
5. [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
Now, let's calculate the ratio between successive terms to check for a consistent pattern:
1. The ratio of the second term to the first term:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
2. The ratio of the third term to the second term:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
3. The ratio of the fourth term to the third term:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2
\][/tex]
4. The ratio of the fifth term to the fourth term:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
We noticed that each of these ratios is consistently 2. This indicates that the sequence has a common ratio of 2.
Based on this information, the sequence can be described by the formula [tex]\( f(x+1) = 2 \times f(x) \)[/tex]. However, let's consider the provided options:
- [tex]\( f(x+1) = -2 f(x) \)[/tex]
- [tex]\( f(x+1) = -\frac{1}{2} f(x) \)[/tex]
- [tex]\( f(x+1) = \frac{1}{2} f(x) \)[/tex]
Since the observed ratio is 2, but negative ([tex]\( -2 \)[/tex]) due to the alternating pattern of signs, the correct formula that fits the sequence is:
[tex]\[
f(x+1) = -2 f(x)
\][/tex]
This formula indicates that each term is multiplied by [tex]\(-2\)[/tex] to yield the subsequent term, reflecting both the magnitude and the sign change observed in the sequence.
Let's first convert the sequence terms into improper fractions for easier handling:
1. [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
2. [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
3. [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
4. [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
5. [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
Now, let's calculate the ratio between successive terms to check for a consistent pattern:
1. The ratio of the second term to the first term:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
2. The ratio of the third term to the second term:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
3. The ratio of the fourth term to the third term:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2
\][/tex]
4. The ratio of the fifth term to the fourth term:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
We noticed that each of these ratios is consistently 2. This indicates that the sequence has a common ratio of 2.
Based on this information, the sequence can be described by the formula [tex]\( f(x+1) = 2 \times f(x) \)[/tex]. However, let's consider the provided options:
- [tex]\( f(x+1) = -2 f(x) \)[/tex]
- [tex]\( f(x+1) = -\frac{1}{2} f(x) \)[/tex]
- [tex]\( f(x+1) = \frac{1}{2} f(x) \)[/tex]
Since the observed ratio is 2, but negative ([tex]\( -2 \)[/tex]) due to the alternating pattern of signs, the correct formula that fits the sequence is:
[tex]\[
f(x+1) = -2 f(x)
\][/tex]
This formula indicates that each term is multiplied by [tex]\(-2\)[/tex] to yield the subsequent term, reflecting both the magnitude and the sign change observed in the sequence.
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