Question

On a graph there is a straight line going from below zero to 30 with points of (9,13), (1,20) and (2, 27)
The function of the arithmetic sequence is

Answer 1

The points are (0,13), (1,20) and (2,27)

Answer 2

To clarify, an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. However, based on your description, it seems you're discussing points on a line, which suggests a linear function, rather than an arithmetic sequence.

The points you've listed — (9, 13), (1, 20), and (2, 27) — suggest that we want to find the equation of a linear function (a line) that might pass through these points, or at least to understand the relationship among them.

### Steps to find the equation of the line:

1. **Find the slope (m)**:
The slope \( m \) between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the slope using points (1, 20) and (2, 27):
\[
m = \frac{27 - 20}{2 - 1} = \frac{7}{1} = 7
\]

Now let's calculate the slope between (1, 20) and (9, 13):
\[
m = \frac{13 - 20}{9 - 1} = \frac{-7}{8}
\]

We can see that the slope between different pairs of points is different, indicating that these points do not lie on the same straight line.

### Finding the linear equations for specific pairs:

2. **Equation of the line through (1, 20) and (2, 27)**:
Using the slope \( m = 7 \) and point \( (1, 20) \):
The equation of the line in point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
Plugging in \( m = 7 \), \( x_1 = 1 \), and \( y_1 = 20 \):
\[
y - 20 = 7(x - 1) \\
y - 20 = 7x - 7 \\
y = 7x + 13
\]

3. **Equation of the line through (2, 27) and (9, 13)**:
Using the slope calculated earlier \( m = \frac{-7}{8} \) and point \( (2, 27) \):
\[
y - 27 = \frac{-7}{8}(x - 2) \\
y - 27 = \frac{-7}{8}x + \frac{14}{8} \\
y = \frac{-7}{8}x + \frac{14}{8} + 27 = \frac{-7}{8}x + \frac{14}{8} + \frac{216}{8} = \frac{-7}{8}x + \frac{230}{8}
\]

### Conclusion:
The points (9, 13), (1, 20), and (2, 27) do not all lie on the same straight line. Instead, we calculated:

- The line through points (1, 20) and (2, 27) approximately is \( y = 7x + 13 \)
- The line through points (2, 27) and (9, 13) is \( y = \frac{-7}{8}x + \frac{230}{8} \)

If the intention was to find an arithmetic sequence rather than a linear function, please provide additional information for further clarification!

Answer 3

Thank you for the clarification. Let's analyze the points \( (0, 13) \), \( (1, 20) \), and \( (2, 27) \).

### Step 1: Determine if the points form an arithmetic sequence

To see if the \(y\)-values form an arithmetic sequence, we can calculate the differences between consecutive \(y\)-values:

- Difference between \(y(1)\) and \(y(0)\):
\[
20 - 13 = 7
\]

- Difference between \(y(2)\) and \(y(1)\):
\[
27 - 20 = 7
\]

Since the differences are constant (both equal to 7), the \(y\)-values do indeed form an arithmetic sequence.

### Step 2: Arithmetic Sequence

The \(y\)-values can be expressed in the form of an arithmetic sequence:
- First term \( a = 13 \) (when \( x = 0 \))
- Common difference \( d = 7 \)

The general term for an arithmetic sequence is given by:
\[
a_n = a + (n - 1) \cdot d
\]

Since \(x\) starts at 0:

For \(x = 0\):
\[
y(0) = 13 + (0) \cdot 7 = 13
\]

For \(x = 1\):
\[
y(1) = 13 + (1) \cdot 7 = 13 + 7 = 20
\]

For \(x = 2\):
\[
y(2) = 13 + (2) \cdot 7 = 13 + 14 = 27
\]

### Step 3: Express as a function

You can express this relationship as a linear function of the form:
\[
y = mx + b
\]

Using the points \( (0, 13) \):

1. **Slope (m)**:
From the differences calculated, we already have \( m = 7 \).

2. **Y-Intercept (b)**:
Since at \( x = 0 \), \( y = 13 \), we have:
\[
b = 13
\]

Thus, the equation for the linear function is:
\[
y = 7x + 13
\]

### Conclusion

The points \( (0, 13) \), \( (1, 20) \), and \( (2, 27) \) indeed form an arithmetic sequence with a first term of \( 13 \) and a common difference of \( 7 \). Additionally, they can be expressed by the linear function:
\[
y = 7x + 13
\]