To derive an explicit formula for the \( n \)-th term \( a_n \) of the sequence 25, 20, 15, ..., we first identify the pattern in the sequence.
Observing the terms:
- The first term \( a_1 = 25 \)
- The second term \( a_2 = 20 \)
- The third term \( a_3 = 15 \)
We can see that each term in the sequence decreases by 5. Hence, we have a linear sequence where each term is generated by subtracting a constant difference of 5 from the previous term.
The general formula for the \( n \)-th term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n - 1) \cdot d \] where \( a_1 \) is the first term and \( d \) is the common difference.
In this case:
- \( a_1 = 25 \)
- \( d = -5 \) (because the sequence is decreasing)
Substituting these values into the formula gives us: \[ a_n = 25 + (n - 1)(-5) \] Simplifying this expression: \[ a_n = 25 - 5(n - 1) \] \[ a_n = 25 - 5n + 5 \] \[ a_n = 30 - 5n \]
Thus, the explicit formula for the \( n \)-th term of the sequence is: \[ \boxed{30 - 5n} \]
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