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Question content area topPart 1The 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?200​, 209​, 218​, 227​, ...

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The bids in the online auction form an arithmetic sequence.

To find the rule for the sequence, we need to identify the first term \( a \) and the common difference \( d \).

  1. First term \( a \): The first term of the sequence is \( a = 200 \).

  2. Common difference \( d \): We can find the common difference by subtracting the first term from the second term: \[ d = 209 - 200 = 9 \]

Now that we have \( a = 200 \) and \( d = 9 \), we can write the general rule for the \( n \)-th term of an arithmetic sequence: \[ a_n = a + (n - 1) d \]

Substituting in the values we have: \[ a_n = 200 + (n - 1) \cdot 9 \]

This simplifies to: \[ a_n = 200 + 9n - 9 = 9n + 191 \]

Next, to find the tenth bid, we substitute \( n = 10 \) into the formula: \[ a_{10} = 9(10) + 191 \] \[ a_{10} = 90 + 191 \] \[ a_{10} = 281 \]

Thus, the rule for the arithmetic sequence is: \[ a_n = 9n + 191 \]

And the tenth bid is: \[ \text{Tenth bid} = 281 \]