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Answer :
To determine which formula describes the sequence:
```
-2 2/3, -5 1/3, -10 2/3, -21 1/3, -42 2/3, ...
```
we should check if the sequence follows a geometric pattern and identify the common ratio between consecutive terms. A geometric sequence has the form:
[tex]\[ a, ar, ar^2, ar^3, \ldots \][/tex]
In a geometric sequence, the ratio [tex]\( r = \frac{a_{n+1}}{a_n} \)[/tex] should be constant for all consecutive terms.
Let's analyze the given sequence:
1. Convert each mixed number to an improper fraction or a decimal for easy multiplication:
- [tex]\(-2 \frac{2}{3} = -\frac{8}{3} = -2.6667\)[/tex]
- [tex]\(-5 \frac{1}{3} = -\frac{16}{3} = -5.3333\)[/tex]
- [tex]\(-10 \frac{2}{3} = -\frac{32}{3} = -10.6667\)[/tex]
- [tex]\(-21 \frac{1}{3} = -\frac{64}{3} = -21.3333\)[/tex]
- [tex]\(-42 \frac{2}{3} = -\frac{128}{3} = -42.6667\)[/tex]
2. Find the common ratio by dividing each term by the previous term:
- From [tex]\(-5 \frac{1}{3}\)[/tex] to [tex]\(-2 \frac{2}{3}\)[/tex]:
[tex]\[ r_1 = \frac{-5 \frac{1}{3}}{-2 \frac{2}{3}} = \frac{-16/3}{-8/3} = 2 \][/tex]
- From [tex]\(-10 \frac{2}{3}\)[/tex] to [tex]\(-5 \frac{1}{3}\)[/tex]:
[tex]\[ r_2 = \frac{-10 \frac{2}{3}}{-5 \frac{1}{3}} = \frac{-32/3}{-16/3} = 2 \][/tex]
- From [tex]\(-21 \frac{1}{3}\)[/tex] to [tex]\(-10 \frac{2}{3}\)[/tex]:
[tex]\[ r_3 = \frac{-21 \frac{1}{3}}{-10 \frac{2}{3}} = \frac{-64/3}{-32/3} = 2 \][/tex]
- From [tex]\(-42 \frac{2}{3}\)[/tex] to [tex]\(-21 \frac{1}{3}\)[/tex]:
[tex]\[ r_4 = \frac{-42 \frac{2}{3}}{-21 \frac{1}{3}} = \frac{-128/3}{-64/3} = 2 \][/tex]
3. All these calculations show the common ratio [tex]\( r = 2 \)[/tex].
Since the ratio between consecutive terms is consistent at 2, the sequence is indeed geometric, and the correct function to describe it is:
[tex]\[ f(x+1) = 2 f(x) \][/tex]
```
-2 2/3, -5 1/3, -10 2/3, -21 1/3, -42 2/3, ...
```
we should check if the sequence follows a geometric pattern and identify the common ratio between consecutive terms. A geometric sequence has the form:
[tex]\[ a, ar, ar^2, ar^3, \ldots \][/tex]
In a geometric sequence, the ratio [tex]\( r = \frac{a_{n+1}}{a_n} \)[/tex] should be constant for all consecutive terms.
Let's analyze the given sequence:
1. Convert each mixed number to an improper fraction or a decimal for easy multiplication:
- [tex]\(-2 \frac{2}{3} = -\frac{8}{3} = -2.6667\)[/tex]
- [tex]\(-5 \frac{1}{3} = -\frac{16}{3} = -5.3333\)[/tex]
- [tex]\(-10 \frac{2}{3} = -\frac{32}{3} = -10.6667\)[/tex]
- [tex]\(-21 \frac{1}{3} = -\frac{64}{3} = -21.3333\)[/tex]
- [tex]\(-42 \frac{2}{3} = -\frac{128}{3} = -42.6667\)[/tex]
2. Find the common ratio by dividing each term by the previous term:
- From [tex]\(-5 \frac{1}{3}\)[/tex] to [tex]\(-2 \frac{2}{3}\)[/tex]:
[tex]\[ r_1 = \frac{-5 \frac{1}{3}}{-2 \frac{2}{3}} = \frac{-16/3}{-8/3} = 2 \][/tex]
- From [tex]\(-10 \frac{2}{3}\)[/tex] to [tex]\(-5 \frac{1}{3}\)[/tex]:
[tex]\[ r_2 = \frac{-10 \frac{2}{3}}{-5 \frac{1}{3}} = \frac{-32/3}{-16/3} = 2 \][/tex]
- From [tex]\(-21 \frac{1}{3}\)[/tex] to [tex]\(-10 \frac{2}{3}\)[/tex]:
[tex]\[ r_3 = \frac{-21 \frac{1}{3}}{-10 \frac{2}{3}} = \frac{-64/3}{-32/3} = 2 \][/tex]
- From [tex]\(-42 \frac{2}{3}\)[/tex] to [tex]\(-21 \frac{1}{3}\)[/tex]:
[tex]\[ r_4 = \frac{-42 \frac{2}{3}}{-21 \frac{1}{3}} = \frac{-128/3}{-64/3} = 2 \][/tex]
3. All these calculations show the common ratio [tex]\( r = 2 \)[/tex].
Since the ratio between consecutive terms is consistent at 2, the sequence is indeed geometric, and the correct function to describe it is:
[tex]\[ f(x+1) = 2 f(x) \][/tex]
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