Which formula can be used to describe the sequence -2 2/3, -5 1/3, -10 2/3, -21 1/3, -42 2/3
The Correct Answer and Explanation is :
The correct answer is: (n = 1, 2, 3, 4, 5)
To describe the sequence (-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}), we can first express these mixed numbers as improper fractions for easier manipulation:
- (-2 \frac{2}{3} = -\frac{8}{3})
- (-5 \frac{1}{3} = -\frac{16}{3})
- (-10 \frac{2}{3} = -\frac{32}{3})
- (-21 \frac{1}{3} = -\frac{64}{3})
- (-42 \frac{2}{3} = -\frac{128}{3})
The sequence in improper fraction form is:
[
-\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}
]
To identify a formula, we can observe that each term appears to be a multiple of (-\frac{8}{3}), multiplied by powers of 2:
- (-\frac{8}{3} = -\frac{8}{3} \cdot 2^0)
- (-\frac{16}{3} = -\frac{8}{3} \cdot 2^1)
- (-\frac{32}{3} = -\frac{8}{3} \cdot 2^2)
- (-\frac{64}{3} = -\frac{8}{3} \cdot 2^3)
- (-\frac{128}{3} = -\frac{8}{3} \cdot 2^4)
This pattern suggests a geometric progression where each term is the previous term multiplied by a common ratio of 2. Thus, the formula for the (n)-th term of the sequence can be expressed as:
[
a_n = -\frac{8}{3} \cdot 2^{n-1}
]
for (n = 1, 2, 3, 4, 5).
Explanation
This formula captures the essence of the sequence’s growth. The initial term, (-\frac{8}{3}), serves as the base value, while the factor (2^{n-1}) accounts for the exponential increase seen in the sequence. Each subsequent term increases in magnitude, reflecting a doubling pattern (as seen in the ratio between consecutive terms).
The negative sign indicates that all terms are negative, while the fractional part ensures that the values stay consistent with the original sequence. The geometric nature of this sequence makes it straightforward to compute future terms by simply applying the formula, allowing for efficient calculation and analysis of the sequence’s behavior.