Question

Given: -8, -11, -14, -17, -20, ...



What is the common difference for this Arithmetic Sequence?


d
=
d=
-3



Write the equation (formula) for the given Arithmetic Sequence.


a
n
=
a
n
​
=




Find the 30th term of this Arithmetic Sequence.


a
30
=
a
30
​
=




Write the equation (slope-intercept form) for the given Arithmetic Sequence.

Answer 1

To analyze the given arithmetic sequence, we first identify the common difference and then formulate the equation for the sequence.

1. **Common Difference (d)**:
The common difference \(d\) can be determined by subtracting any term from the following term.

For example:
\[
d = -11 - (-8) = -3
\]
\[
d = -14 - (-11) = -3
\]
\[
d = -17 - (-14) = -3
\]
This confirms that the common difference is \(d = -3\).

2. **Formula for the Arithmetic Sequence (an)**:
The general formula for the nth term of an arithmetic sequence can be written as:
\[
a_n = a_1 + (n - 1)d
\]
where \(a_1\) is the first term and \(d\) is the common difference.

Here, \(a_1 = -8\) and \(d = -3\). Thus, the formula becomes:
\[
a_n = -8 + (n - 1)(-3)
\]
Simplifying this:
\[
a_n = -8 - 3(n - 1)
\]
\[
a_n = -8 - 3n + 3
\]
\[
a_n = -3n - 5
\]

3. **Finding the 30th Term (a30)**:
We use the formula we derived to find the 30th term:
\[
a_{30} = -3(30) - 5
\]
\[
a_{30} = -90 - 5
\]
\[
a_{30} = -95
\]

4. **Slope-Intercept Form**:
The slope-intercept form of a linear equation is given by \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. In our context, the nth term formula can be reinterpreted as:
\[
a_n = -3n - 5
\]
Here, the slope \(m = -3\) and the y-intercept \(b = -5\).

**To summarize:**
- The common difference \(d = -3\).
- The formula for the sequence:
\[
a_n = -3n - 5
\]
- The 30th term:
\[
a_{30} = -95
\]
- The slope-intercept form of the sequence:
\[
y = -3x - 5
\]