To find the explicit formula for the nth term \( A_n \) of the arithmetic sequence \( 25, 19, 13, 7, \ldots \), we first need to identify the first term and the common difference.
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First term (\( A_1 \)):
- The first term \( A_1 = 25 \).
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Common difference (\( d \)):
- The common difference can be found by subtracting the first term from the second term: \[ d = A_2 - A_1 = 19 - 25 = -6 \]
Now that we have the first term and the common difference, we can use the formula for the nth term of an arithmetic sequence: \[ A_n = A_1 + (n - 1) \cdot d \]
Substituting the values we have: \[ A_n = 25 + (n - 1)(-6) \]
Now, simplify the formula: \[ A_n = 25 - 6(n - 1) \] \[ A_n = 25 - 6n + 6 \] \[ A_n = 31 - 6n \]
Thus, the explicit formula for the nth term \( A_n \) is: \[ A_n = 31 - 6n \]