Answer :
Sure! Let's solve this step-by-step.
First, we need to determine the relationship between the successive terms in 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]
### Step 1: Convert mixed numbers to improper fractions
The sequence terms are mixed numbers, which we need to convert to improper fractions for easier calculation.
1. [tex]\(-2 \frac{2}{3}\)[/tex] is:
[tex]\[
-2 \frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\frac{6}{3} - \frac{2}{3} = -\frac{8}{3}
\][/tex]
2. [tex]\(-5 \frac{1}{3}\)[/tex] is:
[tex]\[
-5 \frac{1}{3} = -\left(5 + \frac{1}{3}\right) = -\frac{15}{3} - \frac{1}{3} = -\frac{16}{3}
\][/tex]
3. [tex]\(-10 \frac{2}{3}\)[/tex] is:
[tex]\[
-10 \frac{2}{3} = -\left(10 + \frac{2}{3}\right) = -\frac{30}{3} - \frac{2}{3} = -\frac{32}{3}
\][/tex]
4. [tex]\(-21 \frac{1}{3}\)[/tex] is:
[tex]\[
-21 \frac{1}{3} = -\left(21 + \frac{1}{3}\right) = -\frac{63}{3} - \frac{1}{3} = -\frac{64}{3}
\][/tex]
5. [tex]\(-42 \frac{2}{3}\)[/tex] is:
[tex]\[
-42 \frac{2}{3} = -\left(42 + \frac{2}{3}\right) = -\frac{126}{3} - \frac{2}{3} = -\frac{128}{3}
\][/tex]
So, the sequence in improper fractions is:
[tex]\[ -\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}, \ldots \][/tex]
### Step 2: Determine the common ratio
The ratio between successive terms is:
[tex]\[
\frac{\text{Next term}}{\text{Previous term}}
\][/tex]
Let's calculate the ratio:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2
\][/tex]
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2
\][/tex]
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = \frac{64}{32} = 2
\][/tex]
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = \frac{128}{64} = 2
\][/tex]
From these calculations, we see that the ratio is consistent between terms, and it is [tex]\(-2\)[/tex].
### Step 3: Identify the formula
The common ratio is [tex]\(-2\)[/tex], which matches the first option in the given choices:
[tex]\[ f(x+1) = -2 f(x) \][/tex]
Therefore, the formula that can be used to describe the sequence is:
[tex]\[ f(x+1) = -2 f(x) \][/tex]
First, we need to determine the relationship between the successive terms in 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]
### Step 1: Convert mixed numbers to improper fractions
The sequence terms are mixed numbers, which we need to convert to improper fractions for easier calculation.
1. [tex]\(-2 \frac{2}{3}\)[/tex] is:
[tex]\[
-2 \frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\frac{6}{3} - \frac{2}{3} = -\frac{8}{3}
\][/tex]
2. [tex]\(-5 \frac{1}{3}\)[/tex] is:
[tex]\[
-5 \frac{1}{3} = -\left(5 + \frac{1}{3}\right) = -\frac{15}{3} - \frac{1}{3} = -\frac{16}{3}
\][/tex]
3. [tex]\(-10 \frac{2}{3}\)[/tex] is:
[tex]\[
-10 \frac{2}{3} = -\left(10 + \frac{2}{3}\right) = -\frac{30}{3} - \frac{2}{3} = -\frac{32}{3}
\][/tex]
4. [tex]\(-21 \frac{1}{3}\)[/tex] is:
[tex]\[
-21 \frac{1}{3} = -\left(21 + \frac{1}{3}\right) = -\frac{63}{3} - \frac{1}{3} = -\frac{64}{3}
\][/tex]
5. [tex]\(-42 \frac{2}{3}\)[/tex] is:
[tex]\[
-42 \frac{2}{3} = -\left(42 + \frac{2}{3}\right) = -\frac{126}{3} - \frac{2}{3} = -\frac{128}{3}
\][/tex]
So, the sequence in improper fractions is:
[tex]\[ -\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}, \ldots \][/tex]
### Step 2: Determine the common ratio
The ratio between successive terms is:
[tex]\[
\frac{\text{Next term}}{\text{Previous term}}
\][/tex]
Let's calculate the ratio:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2
\][/tex]
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2
\][/tex]
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = \frac{64}{32} = 2
\][/tex]
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = \frac{128}{64} = 2
\][/tex]
From these calculations, we see that the ratio is consistent between terms, and it is [tex]\(-2\)[/tex].
### Step 3: Identify the formula
The common ratio is [tex]\(-2\)[/tex], which matches the first option in the given choices:
[tex]\[ f(x+1) = -2 f(x) \][/tex]
Therefore, the formula that can be used to describe the sequence is:
[tex]\[ f(x+1) = -2 f(x) \][/tex]
