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 \(304\).