Question

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​, ...

Answer 1

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:
\[
\text{Tenth bid} = 281
\]