Question

What represents the recursive formula for the arithmetic sequence an=7-3(n-1)

Answer 1

`\left\lbrace\begin{aligned}& a_(1) = 7 \\ & a_(n+1) = a_(n) - 3 && \text{for $n \ge 1$}\end{aligned}\right.`.

Explanation:

An arithmetic sequence could be defined using only two pieces of information:

• The first term of this sequence,
  `a_(1)`.
• The common difference between two consecutive terms of this sequence,
  `d`.

The arithmetic sequence in this question is given in the explicit form,
`a_(n) = 7 - 3\, (n - 1)`, which is equivalent to
`a_(n) = 7 + (-3)\, (n - 1)`. In the explicit form,
`a_(1)` and
`d` are described in the same equation:

`\text{$a_(n) = a_(1) + d\, (n - 1)$ for $n \ge 1$}`.

Compare
`a_(n) = a_(1) + d\, (n - 1)` with
`a_(n) = 7 + (-3)\, (n - 1)`:
`a_(1) = 7` whereas
`d = -3`.

When an arithmetic sequence is given in the recursive form,
`a_(1)` and
`d` are specified in two separate equations:

`\left\lbrace\begin{aligned}& a_(1) = a_(1) \\ & a_(n+1) = a_(n) + d && \text{for $n \ge 1$}\end{aligned}\right.`.

In this question, it was found that
`a_(1) = 7` whereas
`d = -3`. Hence, the corresponding recursive formula for this arithmetic sequence would be:

`\left\lbrace\begin{aligned}& a_(1) = 7 \\ & a_(n+1) = a_(n) + (-3) && \text{for $n \ge 1$}\end{aligned}\right.`.