Answer:
[tex]a_n=-3n+11[/tex]
Step-by-step explanation:
We are given the values of two terms in this arithmetic sequence: [tex]a_{17}=-40[/tex] and [tex]a_{28}=-73[/tex]. We want to find the recursive formula of this sequence, which will be in the form [tex]a_n=a_1+d(n-1)[/tex], where [tex]a_1[/tex] is the first term and d is the common difference.
Here, we can pretend that [tex]a_{17}[/tex] will replace the [tex]a_1[/tex] term, while [tex]a_{28}[/tex] replaces the [tex]a_n[/tex] term. This way, n becomes 28 and 1 becomes 17. Now, we can write:
[tex]a_n=a_1+d(n-1)[/tex]
[tex]a_{28}=a_{17}+d(28-17)[/tex]
Substitute in the values we know:
[tex]a_{28}=a_{17}+d(28-17)[/tex]
[tex]-73=-40+d(28-17)[/tex]
Solve for d:
[tex]-73=-40+d(28-17)[/tex]
[tex]-73=-40+d(11)[/tex]
11d = -33
d = -3
Now, we need to find our first term. We can do this by replacing [tex]a_n[/tex] with [tex]a_{28}[/tex] again, but this time, we're actually going to use [tex]a_1[/tex]:
[tex]a_{28}=a_1+d(28-1)[/tex]
Plug in the values we know:
[tex]a_{28}=a_1+d(28-1)[/tex]
[tex]-73=a_1-3(27)[/tex]
Solve for [tex]a_1[/tex]:
-73 = [tex]a_1[/tex] - 81
[tex]a_1[/tex] = 8
Put these altogether:
[tex]a_n=8-3(n-1)=-3n+11[/tex]
Thus, the recursive formula is [tex]a_n=-3n+11[/tex].
