The sequence given is:
-1, -3, -9, ...
To find a pattern, let's observe the relationship between the terms.
- The first term \( a_1 = -1 \).
- The second term \( a_2 = -3 \).
- The third term \( a_3 = -9 \).
We can see that the terms are decreasing in a specific manner. Let's analyze the possibility of relating these terms to powers or multiplication.
If we look closely:
- The second term \(-3\) can be expressed as \(-1 \times 3\).
- The third term \(-9\) can be expressed as \(-3 \times 3\).
This suggests that each term can be derived from the previous term by multiplying it by \(3\).
To formalize this observation, we can define the \(n\)-th term of the sequence using a recursive approach; however, since we need a direct formula, let's analyze:
The terms can also be written:
- \( a_1 = -1 = -1 \times 3^0 \)
- \( a_2 = -3 = -1 \times 3^1 \)
- \( a_3 = -9 = -1 \times 3^2 \)
From this, we can deduce that the \(n\)-th term can be expressed as: \[ a_n = -1 \times 3^{n-1} \] This can be simplified to: \[ a_n = -3^{n-1} \]
Thus, the equation you requested, where \(n\) represents the position of a term (with \(n=1\) for the first term), is:
\[ a_n = -3^{n-1} \]
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