For an arithmetic sequence the n th term ( explicit formula ) is
[tex]a_{n}[/tex] = [tex]a_{1}[/tex] + (n - 1 )d
where d is the common difference and [tex]a_{1}[/tex] the first term
here d = 4 - 1 = 1 - (- 2) = - 2 - (- 5) = 3 and [tex]a_{1}[/tex] = - 5, hence
[tex]a_{n}[/tex] = - 5 + 3(n - 1) = - 5 + 3n - 3 = 3n - 8 ← explicit formula
A recursive formula allows us to find the next term in the sequence from the previous term, hence
[tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] + 3, with [tex]a_{1}[/tex] = - 5
