Given:
common difference (d) = 7
6th term = 53
n = 6
Find: 89th term where n = 89
Solution:
To determine the 89th term, let's first determine the first term using the recursive formula for an arithmetic sequence.
[tex]a_1=a_n-(n-1)(d)[/tex]
Let's use the given 6th term in which n = 6 and a₆ = 53. Use d = 7 for the common difference. Let's plug this into the recursive formula above.
[tex]\begin{gathered} a_1=53-(6-1)(7) \\ a_1=53-(5)(7) \\ a_1=53-35 \\ a_1=18 \end{gathered}[/tex]
Therefore, the first term is 18.
Now, let's use the recursive formula again to determine the 89th term.
This time, we will use n = 89, d = 7, and a₁ = 18.
[tex]a_n=a_1+(n-1)(d)[/tex][tex]\begin{gathered} a_{89}=18+(89-1)(7) \\ a_{89}=18+(88)(7) \\ a_{89}=18+616 \\ a_{89}=634 \end{gathered}[/tex]
Therefore, the 89th term in the sequence is 634.
