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 \( a_n \).
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Identify the sequence: The sequence starts with 8, and each subsequent term seems to alternate in sign and doubles in absolute value:
- \( a_1 = 8 \)
- \( a_2 = -16 \)
- \( a_3 = 32 \)
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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.
- The terms can be described as:
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Generalize the expression:
- We can write the n-th term as:
- \( a_n = 8 \cdot (-2)^{n-1} \)
- We can write the n-th term as:
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 \( n \).
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