Question

How does the common difference of an arithmetic sequence relate to the sequence’s corresponding linear function?
A. It is the first term of the linear function.
B. It is the slope of the linear function.
C. It is the common ratio of the linear function.
D. It is the y-intercept of the linear function.

Which represents the explicit formula for the arithmetic sequence An=7−3(n−1) in function form?
A. f(n)=3n−6
B. f(n)=−3n−6
C. f(n)=3n−10

Which is never visible on the graph of a sequence that starts with n = 1?
A. x−intercept
B. y−intercept
C. common difference
D. the first term

Answer 1

Answer:

1) OPTION B.

2) $f(n)=-3n+10$

3) OPTION B.

Step-by-step explanation:

1) In linear functions the variables change at a constant rate (which is the slope of the line).

By definition, the difference  between consecutive terms in an arithmetic sequence is constant; this is called the "Common difference" (Represented  with $d$).

Therefore, an arithmetic sequence is a linear function, where the Common difference is the slope.

2) The explicit formula for an arithmetic sequence has this form:

$a_n= a_1 + d (n - 1)$

Where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number and $d$ is the common difference.

In function notation, this is:

$f(n)=f(1)+d(n-1)$

Where $f(n)$ is the nth term, $f(1)$ is the first term, $n$ is the term number and $d$ is the common difference.

Given the explicit formula for the arithmetic sequence provided in the exercise, we can identify that:

$f(1)=a_1=7\\\\d=-3$

Therefore, written in function form, this is:

$f(n)=7-3(n-1)\\\\f(n)=7-3n+3\\\\f(n)=-3n+10$

3) By definition, the line intersects the y-axis when the x-coordinate is 0.

Therefore, if the graph of a sequence starts with $n=1$, the y-intercept is not visible.