To identify the recursive formula for the given sequence [tex]\(900, 850, 800, 750, \ldots\)[/tex], we need to observe the pattern in the sequence and determine how each term relates to the previous term.
1. Identify the Pattern:
- We start with the first term [tex]\(a_1 = 900\)[/tex].
- The second term [tex]\(a_2 = 850\)[/tex].
- The third term [tex]\(a_3 = 800\)[/tex], and so on.
2. Determine the Difference:
- To find the common difference ([tex]\(d\)[/tex]), we subtract the second term from the first term and check if the difference remains constant throughout the sequence.
[tex]\[
d = 850 - 900 = -50
\][/tex]
[tex]\[
d = 800 - 850 = -50
\][/tex]
[tex]\[
d = 750 - 800 = -50
\][/tex]
The common difference ([tex]\(d\)[/tex]) is [tex]\(-50\)[/tex], confirming that this is an arithmetic sequence where each term decreases by 50.
3. Form the Recursive Formula:
- For an arithmetic sequence, the recursive formula generally has the form:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
- Here, the common difference [tex]\(d\)[/tex] is [tex]\(-50\)[/tex]. Therefore,
[tex]\[
a_n = a_{n-1} + (-50)
\][/tex]
- Simplifying this, we get:
[tex]\[
a_n = a_{n-1} - 50
\][/tex]
4. State the Recursive Formula:
- The recursive formula for the given sequence [tex]\(900, 850, 800, 750, \ldots\)[/tex] is:
[tex]\[
a_n = a_{n-1} - 50
\][/tex]
This formula indicates that to find any term in the sequence, you subtract 50 from the previous term.
