Answer :
To solve this problem, we need to identify a pattern or formula that describes the given sequence:
The sequence provided is:
[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
First, let's convert each term in the sequence to an improper fraction for easier manipulation:
- [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
Step 2: Determine the Ratio Between Terms
Next, we examine the ratio between consecutive terms to determine if there is a common ratio, suggesting a geometric sequence:
- The ratio between the first and second term:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- The ratio between the second and third term:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
- The ratio between the third and fourth term:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2
\][/tex]
- The ratio between the fourth and fifth term:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
Step 3: Conclude the Pattern
Since the ratio between consecutive terms is consistently [tex]\(2\)[/tex], we identify this sequence as a geometric sequence with a common ratio of [tex]\(2\)[/tex].
Conclusion:
The formula that describes this sequence is:
[tex]\[
f(x+1) = 2f(x)
\][/tex]
This means each term in the sequence is obtained by multiplying the previous term by 2. Thus, the correct formula is [tex]\(f(x+1) = 2 f(x)\)[/tex].
The sequence provided is:
[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
First, let's convert each term in the sequence to an improper fraction for easier manipulation:
- [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
Step 2: Determine the Ratio Between Terms
Next, we examine the ratio between consecutive terms to determine if there is a common ratio, suggesting a geometric sequence:
- The ratio between the first and second term:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- The ratio between the second and third term:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
- The ratio between the third and fourth term:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2
\][/tex]
- The ratio between the fourth and fifth term:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
Step 3: Conclude the Pattern
Since the ratio between consecutive terms is consistently [tex]\(2\)[/tex], we identify this sequence as a geometric sequence with a common ratio of [tex]\(2\)[/tex].
Conclusion:
The formula that describes this sequence is:
[tex]\[
f(x+1) = 2f(x)
\][/tex]
This means each term in the sequence is obtained by multiplying the previous term by 2. Thus, the correct formula is [tex]\(f(x+1) = 2 f(x)\)[/tex].
