To analyze the given arithmetic sequence, we first identify the common difference and then formulate the equation for the sequence.
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Common Difference (d): The common difference \(d\) can be determined by subtracting any term from the following term.
For example: \[ d = -11 - (-8) = -3 \] \[ d = -14 - (-11) = -3 \] \[ d = -17 - (-14) = -3 \] This confirms that the common difference is \(d = -3\).
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Formula for the Arithmetic Sequence (an): The general formula for the nth term of an arithmetic sequence can be written as: \[ a_n = a_1 + (n - 1)d \] where \(a_1\) is the first term and \(d\) is the common difference.
Here, \(a_1 = -8\) and \(d = -3\). Thus, the formula becomes: \[ a_n = -8 + (n - 1)(-3) \] Simplifying this: \[ a_n = -8 - 3(n - 1) \] \[ a_n = -8 - 3n + 3 \] \[ a_n = -3n - 5 \]
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Finding the 30th Term (a30): We use the formula we derived to find the 30th term: \[ a_{30} = -3(30) - 5 \] \[ a_{30} = -90 - 5 \] \[ a_{30} = -95 \]
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Slope-Intercept Form: The slope-intercept form of a linear equation is given by \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. In our context, the nth term formula can be reinterpreted as: \[ a_n = -3n - 5 \] Here, the slope \(m = -3\) and the y-intercept \(b = -5\).
To summarize:
- The common difference \(d = -3\).
- The formula for the sequence: \[ a_n = -3n - 5 \]
- The 30th term: \[ a_{30} = -95 \]
- The slope-intercept form of the sequence: \[ y = -3x - 5 \]
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