Answer :
Sure! Let's analyze the sequence step-by-step to determine the formula that describes it:
1. List the sequence: The sequence is given as:
[tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots\)[/tex].
2. Convert mixed numbers into improper fractions for easier computation:
- [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]
3. Identify the pattern in the numerators:
- The numerators are [tex]\(-8, -16, -32, -64, -128\)[/tex].
4. Determine the relation between consecutive terms:
- Each numerator is [tex]\(2\)[/tex] times the previous numerator:
- [tex]\(-16 = 2 \times (-8)\)[/tex]
- [tex]\(-32 = 2 \times (-16)\)[/tex]
- [tex]\(-64 = 2 \times (-32)\)[/tex]
- [tex]\(-128 = 2 \times (-64)\)[/tex]
5. Generalize the pattern:
- Since each term is being multiplied by [tex]\(2\)[/tex] from one term to the next, this sequence demonstrates a geometric pattern with a common ratio. However, due to alternating signs, the ratio between the terms is actually [tex]\(-2\)[/tex].
6. Find the correct formula:
- A rule for a geometric sequence is usually of the form [tex]\( f(x+1) = r \cdot f(x) \)[/tex], where [tex]\(r\)[/tex] is the common ratio.
- Here, since each term is the previous term multiplied by [tex]\(-2\)[/tex], the sequence is described by the formula:
[tex]\[
f(x+1) = -2 \cdot f(x)
\][/tex]
Therefore, the correct formula to describe this sequence is [tex]\( f(x+1) = -2 \cdot f(x) \)[/tex].
1. List the sequence: The sequence is given as:
[tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots\)[/tex].
2. Convert mixed numbers into improper fractions for easier computation:
- [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]
3. Identify the pattern in the numerators:
- The numerators are [tex]\(-8, -16, -32, -64, -128\)[/tex].
4. Determine the relation between consecutive terms:
- Each numerator is [tex]\(2\)[/tex] times the previous numerator:
- [tex]\(-16 = 2 \times (-8)\)[/tex]
- [tex]\(-32 = 2 \times (-16)\)[/tex]
- [tex]\(-64 = 2 \times (-32)\)[/tex]
- [tex]\(-128 = 2 \times (-64)\)[/tex]
5. Generalize the pattern:
- Since each term is being multiplied by [tex]\(2\)[/tex] from one term to the next, this sequence demonstrates a geometric pattern with a common ratio. However, due to alternating signs, the ratio between the terms is actually [tex]\(-2\)[/tex].
6. Find the correct formula:
- A rule for a geometric sequence is usually of the form [tex]\( f(x+1) = r \cdot f(x) \)[/tex], where [tex]\(r\)[/tex] is the common ratio.
- Here, since each term is the previous term multiplied by [tex]\(-2\)[/tex], the sequence is described by the formula:
[tex]\[
f(x+1) = -2 \cdot f(x)
\][/tex]
Therefore, the correct formula to describe this sequence is [tex]\( f(x+1) = -2 \cdot f(x) \)[/tex].
