To find the explicit and recursive equations for the arithmetic sequence given \( a_1 = -1 \) (the first term) and \( d = -10 \) (the common difference), we can proceed as follows:
Explicit Equation
The explicit formula for the \( n \)-th term of an arithmetic sequence is given by:
\[ a_n = a_1 + (n - 1) \cdot d \]
Substituting the values for \( a_1 \) and \( d \):
\[ a_n = -1 + (n - 1) \cdot (-10) \]
Simplifying this:
\[ a_n = -1 - 10(n - 1) \] \[ a_n = -1 - 10n + 10 \] \[ a_n = 9 - 10n \]
Recursive Equation
The recursive formula for an arithmetic sequence is given by:
\[ a_n = a_{n-1} + d \]
For this sequence, we can define it as:
\[ a_1 = -1 \] \[ a_n = a_{n-1} - 10 \quad \text{for } n > 1 \]
Finding the 25th Term
Now, we need to find \( a_{25} \).
Using the explicit formula:
\[ a_{25} = 9 - 10(25) \] \[ a_{25} = 9 - 250 \] \[ a_{25} = -241 \]
Summary
- Explicit equation: \( a_n = 9 - 10n \)
- Recursive equation:
- \( a_1 = -1 \)
- \( a_n = a_{n-1} - 10 \) for \( n > 1 \)
- The 25th term \( a_{25} \) is \(-241\).