Question

Given an arithmetic sequence in the table below, create the explicit formula and list any restrictions to the domain. (2 points) n an 1 40 2 47 3 54 an = 40 + 7(n − 1) where n ≥ 40 an = 40 + 7(n − 1) where n ≥ 1 an = 40 − 7(n − 1) where n ≥ 40 an = 40 − 7(n − 1) where n ≥ 1

Answer 1

an = 40 +7(n-1) where n is greater than or equal to 1

Answer 2

An explicit formula for the arithmetic sequence is $a_n=40+7(n-1)$ and the domain is $\mathbb{N}$ or $\{n \text{ }|\text{ }n\in \mathbb{Z}\text{ and } n\ge 1\}$.

The sequence is $40,47,54,...$ with $a_1=40$

The general formula for an arithmetic sequence is

$a_n=a+(n-1)d$

in our case,

$a_n=40+(n-1)d$

Since subsequent terms differ by steps of $7$, $d=7$. So, we now have

$a_n=40+7(n-1)$

But, what values of $n$ will make $a_n$ valid? or, what is the domain of $a_n$?

• $n$ must have integral values. Using fractional values will give us elements not in the sequence.
• $n$ cannot be zero, or negative either. The first term is gotten from $n=1$. Other terms are gotten from $n=2,3,4,...$ .Values of $n<1$ produce elements not in the sequence.

The domain of $n$ is $\mathbb{N}$ or $\{n \text{ }|\text{ }n\in \mathbb{Z}\text{ and } n\ge 1\}$.

Therefore,

The formula is $a_n=40+7(n-1)$ and the domain is $\mathbb{N}$ or $\{n \text{ }|\text{ }n\in \mathbb{Z}\text{ and } n\ge 1\}$.

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