Question

(0, 13), (1, 20), and (2, 27) find the arthmetic sequence function of the following points

Answer 1

To find the arithmetic sequence function that represents the given points \((0, 13)\), \((1, 20)\), and \((2, 27)\), we can first observe the outputs (y-values) of the points:

- For \(x = 0\), \(y = 13\)
- For \(x = 1\), \(y = 20\)
- For \(x = 2\), \(y = 27\)

Next, we can check the differences between successive y-values to confirm that this is indeed an arithmetic sequence:

- The difference between \(y\) values at \(x = 1\) and \(x = 0\) is:
\[
20 - 13 = 7
\]

- The difference between \(y\) values at \(x = 2\) and \(x = 1\) is:
\[
27 - 20 = 7
\]

Since the difference is constant (\(7\)), we can establish that this represents an arithmetic sequence.

To express this as a function, \(y\) can be represented in terms of \(x\) in the form of:

\[
y = mx + b
\]

where \(m\) is the common difference and \(b\) is the y-intercept. From our earlier calculation, we found \(m = 7\).

Now we need to find the y-intercept \(b\). We know that when \(x = 0\), \(y = 13\), therefore:

\[
y = 7(0) + b \implies b = 13
\]

Putting this together, we get the function:

\[
y = 7x + 13
\]

Thus, the function representing the arithmetic sequence for the given points is:

\[
\boxed{y = 7x + 13}
\]