Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


8 , –16 , 32 , ...


Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

To find an equation that describes the sequence 8, -16, 32, ..., we first need to analyze the pattern of the sequence. Let's denote the n-th term of the sequence as \( a_n \).

1. **Identify the sequence**: The sequence starts with 8, and each subsequent term seems to alternate in sign and doubles in absolute value:
- \( a_1 = 8 \)
- \( a_2 = -16 \)
- \( a_3 = 32 \)

2. **Establish the pattern**:
- The terms can be described as:
- For odd n (like \( n = 1, 3, ... \)), the terms are positive and appear to be \( 8 \cdot 2^{k} \) where \( k \) is the count of the odd indexed term.
- For even n (like \( n = 2, 4, ... \)), the terms are negative and appear to be \( -8 \cdot 2^{k} \) where \( k \) is the count of the even indexed term.

3. **Generalize the expression**:
- We can write the n-th term as:
- \( a_n = 8 \cdot (-2)^{n-1} \)

This formula accounts for the alternation in sign and the doubling of the absolute values. Thus, we can represent the nth term of the sequence as follows:

\[
a_n = 8 \cdot (-2)^{n-1}
\]

This expression uses integers and can be understood for any integer value of \( n \).