Respuesta :
An explicit formula for the arithmetic sequence is [tex]a_n=40+7(n-1)[/tex] and the domain is [tex]\mathbb{N}[/tex] or [tex]\{n \text{ }|\text{ }n\in \mathbb{Z}\text{ and } n\ge 1\}[/tex].
The sequence is [tex]40,47,54,...[/tex] with [tex]a_1=40[/tex]
The general formula for an arithmetic sequence is
[tex]a_n=a+(n-1)d[/tex]
in our case,
[tex]a_n=40+(n-1)d[/tex]
Since subsequent terms differ by steps of [tex]7[/tex], [tex]d=7[/tex]. So, we now have
[tex]a_n=40+7(n-1)[/tex]
But, what values of [tex]n[/tex] will make [tex]a_n[/tex] valid? or, what is the domain of [tex]a_n[/tex]?
- [tex]n[/tex] must have integral values. Using fractional values will give us elements not in the sequence.
- [tex]n[/tex] cannot be zero, or negative either. The first term is gotten from [tex]n=1[/tex]. Other terms are gotten from [tex]n=2,3,4,...[/tex] .Values of [tex]n<1[/tex] produce elements not in the sequence.
The domain of [tex]n[/tex] is [tex]\mathbb{N}[/tex] or [tex]\{n \text{ }|\text{ }n\in \mathbb{Z}\text{ and } n\ge 1\}[/tex].
Therefore,
The formula is [tex]a_n=40+7(n-1)[/tex] and the domain is [tex]\mathbb{N}[/tex] or [tex]\{n \text{ }|\text{ }n\in \mathbb{Z}\text{ and } n\ge 1\}[/tex].
Learn more about arithmetic sequences here: https://brainly.com/question/25715593