Question

The bids in an online auction are represented by the arithmetic sequence shown below. Write a rule to represent the arithmetic sequence. What is the tenth ​bid?205​, 216​, 227​, 238​, ...Question content area bottomPart 1​A(n)equalsenter your response here ​(Type an expression using n for the​ variable.)

Answer 1

In the given sequence, the bids are as follows:

205, 216, 227, 238, ...

To identify the arithmetic sequence, we first need to find the common difference (d).

Calculating the common difference:

- \(216 - 205 = 11\)
- \(227 - 216 = 11\)
- \(238 - 227 = 11\)

So, the common difference \(d\) is \(11\).

The formula for the \(n\)th term of an arithmetic sequence is given by:

\[
A(n) = A(1) + (n - 1) \cdot d
\]

Where:
- \(A(1)\) is the first term of the sequence,
- \(d\) is the common difference,
- \(n\) is the term number.

Given:
- The first term \(A(1)\) is \(205\),
- The common difference \(d\) is \(11\).

Substituting these values into the formula:

\[
A(n) = 205 + (n - 1) \cdot 11
\]

Simplifying this:

\[
A(n) = 205 + 11n - 11
\]

\[
A(n) = 11n + 194
\]

So the rule to represent the arithmetic sequence is:

\[
A(n) = 11n + 194
\]

Now, to find the tenth bid (\(A(10)\)):

\[
A(10) = 11(10) + 194
\]

Calculating:

\[
A(10) = 110 + 194 = 304
\]

Thus, the tenth bid is \(304\).

Final answers:
- The rule for the arithmetic sequence is \(A(n) = 11n + 194\).
- The tenth bid is \(304\).