Answer :
To find the formula that describes the sequence, we need to identify the pattern or relationship between the terms of the sequence. Here is a step-by-step solution to determine this:
1. Examine the Sequence:
The given sequence is:
[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 Terms to Improper Fractions:
To work with these numbers more easily, let's convert them to improper fractions:
- [tex]\(-2 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{8}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{16}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{32}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{64}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{128}{3}\)[/tex]
3. Calculate the Ratio:
We need to find out how each term is related to the previous term by checking if they follow some multiplication pattern. Let's compute the ratio from one term to the next:
[tex]\[
\text{Ratio} = \frac{\text{Next Term}}{\text{Previous Term}}
\][/tex]
- For the first two terms:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- For the next pair:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
- This pattern continues for all pairs, confirming a common ratio of 2.
4. Understand the Pattern:
Since each term is multiplied by 2 to get the next term, this is indicative of a geometric sequence with a ratio of 2.
5. Write the Formula:
The relation between consecutive terms can be described by the following formula:
[tex]\[
f(x+1) = 2 \cdot f(x)
\][/tex]
6. Select the Correct Formula:
Based on our calculation and analysis, the formula that fits the pattern of this sequence is:
[tex]\[
f(x+1) = 2 \cdot f(x)
\][/tex]
Thus, the correct formula that describes this sequence is [tex]\( f(x+1) = 2 f(x) \)[/tex], matching the characteristic of this geometric sequence.
1. Examine the Sequence:
The given sequence is:
[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 Terms to Improper Fractions:
To work with these numbers more easily, let's convert them to improper fractions:
- [tex]\(-2 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{8}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{16}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{32}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{64}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{128}{3}\)[/tex]
3. Calculate the Ratio:
We need to find out how each term is related to the previous term by checking if they follow some multiplication pattern. Let's compute the ratio from one term to the next:
[tex]\[
\text{Ratio} = \frac{\text{Next Term}}{\text{Previous Term}}
\][/tex]
- For the first two terms:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- For the next pair:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
- This pattern continues for all pairs, confirming a common ratio of 2.
4. Understand the Pattern:
Since each term is multiplied by 2 to get the next term, this is indicative of a geometric sequence with a ratio of 2.
5. Write the Formula:
The relation between consecutive terms can be described by the following formula:
[tex]\[
f(x+1) = 2 \cdot f(x)
\][/tex]
6. Select the Correct Formula:
Based on our calculation and analysis, the formula that fits the pattern of this sequence is:
[tex]\[
f(x+1) = 2 \cdot f(x)
\][/tex]
Thus, the correct formula that describes this sequence is [tex]\( f(x+1) = 2 f(x) \)[/tex], matching the characteristic of this geometric sequence.
