Question

Write an explicit equation and recursive equation for each arithmetic sequence. then find the indicated term.
a_(1) =-1, d= -10, a_(25)

Answer 1

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 \(-241\).