College

Given 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]

Which formula can be used to describe the sequence?

A. [tex] f(x+1) = -2 f(x) [/tex]

B. [tex] f(x+1) = -\frac{1}{2} f(x) [/tex]

C. [tex] f(x+1) = \frac{1}{2} f(x) [/tex]

D. [tex] f(x+1) = 2 f(x) [/tex]

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].