Question

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

Answer 1

Answer:

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

Step-by-step 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:

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