Question

Answer each question based on the following arithmetic sequence.

35, 31, 27, 23,...

Part A

For the sequence listed above, what is
a
1
a
1
​
?


Part B

For the sequence listed above, what is
d
d ?



(b)
Part C
For the sequence listed above, what is its explicit equation?

A
an=−4+35(n−1)a_n=-4+35\left(n-1\right)a
n
​
=−4+35(n−1)

B
an=−4+35(n+1)a_n=-4+35(n+1)a
n
​
=−4+35(n+1)

C
an=35+(−4)(n−1)a_n=35+\left(-4\right)(n-1)a
n
​
=35+(−4)(n−1)

D
an=35+(−4)(n+1)a_n=35+\left(-4\right)(n+1)a
n
​
=35+(−4)(n+1)

(c)

Part D
Translate the arithmetic sequence formula to slope-intercept form:


Make sure that you use the proper variables for slope-intercept form.

Answer 1

Let's go through each part of the problem step by step.

### Part A
The first term in the given arithmetic sequence is represented by \( a_1 \). From the sequence provided:

\[
a_1 = 35
\]

### Part B
The common difference \( d \) in an arithmetic sequence is found by subtracting the first term from the second term:

\[
d = a_2 - a_1 = 31 - 35 = -4
\]

### Part C
The explicit formula for an arithmetic sequence can be expressed as:

\[
a_n = a_1 + (n - 1)d
\]

Substituting the values we found:

- \( a_1 = 35 \)
- \( d = -4 \)

Thus, the formula becomes:

\[
a_n = 35 + (n - 1)(-4)
\]

Simplifying this gives:

\[
a_n = 35 - 4(n - 1)
\]

The choices you provided can be reviewed:

- **Option A:** \( a_n = -4 + 35(n - 1) \) (incorrect)
- **Option B:** \( a_n = -4 + 35(n + 1) \) (incorrect)
- **Option C:** \( a_n = 35 + (-4)(n - 1) \) (correct)
- **Option D:** \( a_n = 35 + (-4)(n + 1) \) (incorrect)

The correct answer is **C**:

\[
a_n = 35 + (-4)(n - 1)
\]

### Part D
To convert the formula \( a_n = 35 + (-4)(n - 1) \) into slope-intercept form \( y = mx + b \), let’s assume \( y = a_n \) and \( x = n \).

Starting with:

\[
a_n = 35 - 4(n - 1)
\]

Expanding this gives:

\[
a_n = 35 - 4n + 4
\]
\[
a_n = -4n + 39
\]

So in slope-intercept form, it would be:

\[
y = -4x + 39
\]

Where:

- The slope \( m = -4 \)
- The y-intercept \( b = 39 \)

Thus, the final answer for Part D is:

\[
y = -4x + 39
\]