The arithmetic sequence provided is: [tex]\(15.75, 14.25, 12.75, \ldots\)[/tex].
To find the [tex]\(n\)[/tex]th term of an arithmetic sequence, we typically use the first term [tex]\(a_1\)[/tex] and the common difference [tex]\(d\)[/tex]. In this sequence, the first term is [tex]\(a_1 = 15.75\)[/tex], and the common difference [tex]\(d\)[/tex] is the difference between consecutive terms. By calculating the difference between the first two terms:
[tex]\[ d = 14.25 - 15.75 = -1.5 \][/tex]
Given that the above sequence has [tex]\(a_1 = 15.75\)[/tex] and [tex]\(d = -1.5\)[/tex], the recursive formula for the [tex]\(n\)[/tex]th term [tex]\(a_n\)[/tex] of this arithmetic sequence is:
[tex]\[ a_1 = 15.75 \][/tex]
[tex]\[ a_n = a_{n-1} + (-1.5) \][/tex]
This can be simplified to:
[tex]\[ a_1 = 15.75 \][/tex]
[tex]\[ a_n = a_{n-1} - 1.5 \][/tex]
Thus, the recursive formula is:
[tex]\[
a_1 = 15.75
\][/tex]
[tex]\[
a_n = a_{n-1} - 1.5
\][/tex]
This formula correctly defines the sequence [tex]\(15.75, 14.25, 12.75, \ldots \)[/tex] and can be used to find any term in the sequence recursively.
