Answer:
[tex]\left\lbrace\begin{aligned}& a_{1} = 7 \\ & a_{n+1} = a_{n} - 3 && \text{for $n \ge 1$}\end{aligned}\right.[/tex].
Step-by-step explanation:
An arithmetic sequence could be defined using only two pieces of information:
The arithmetic sequence in this question is given in the explicit form, [tex]a_{n} = 7 - 3\, (n - 1)[/tex], which is equivalent to [tex]a_{n} = 7 + (-3)\, (n - 1)[/tex]. In the explicit form, [tex]a_{1}[/tex] and [tex]d[/tex] are described in the same equation:
[tex]\text{$a_{n} = a_{1} + d\, (n - 1)$ for $n \ge 1$}[/tex].
Compare [tex]a_{n} = a_{1} + d\, (n - 1)[/tex] with [tex]a_{n} = 7 + (-3)\, (n - 1)[/tex]: [tex]a_{1} = 7[/tex] whereas [tex]d = -3[/tex].
When an arithmetic sequence is given in the recursive form, [tex]a_{1}[/tex] and [tex]d[/tex] are specified in two separate equations:
[tex]\left\lbrace\begin{aligned}& a_{1} = a_{1} \\ & a_{n+1} = a_{n} + d && \text{for $n \ge 1$}\end{aligned}\right.[/tex].
In this question, it was found that [tex]a_{1} = 7[/tex] whereas [tex]d = -3[/tex]. Hence, the corresponding recursive formula for this arithmetic sequence would be:
[tex]\left\lbrace\begin{aligned}& a_{1} = 7 \\ & a_{n+1} = a_{n} + (-3) && \text{for $n \ge 1$}\end{aligned}\right.[/tex].
