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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 eleventh ​bid?

187​, 199​, 211​, 223​, ...
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Part 1
​A(n)equals
  
enter your response here ​(Type an expression using n for the​ variable.)

1 answer

To find a rule that represents the arithmetic sequence, we first identify the first term and the common difference.

The first term \( a_1 \) is: \[ a_1 = 187 \]

Next, we find the common difference \( d \): \[ d = 199 - 187 = 12 \] This common difference of \( 12 \) holds for the subsequent terms as well: \[ 211 - 199 = 12, \quad 223 - 211 = 12 \]

The general formula for the \( n \)-th term of an arithmetic sequence is given by: \[ A(n) = a_1 + (n - 1) \cdot d \]

Substituting the known values: \[ A(n) = 187 + (n - 1) \cdot 12 \]

Now we can simplify this expression: \[ A(n) = 187 + 12n - 12 \] \[ A(n) = 12n + 175 \]

Thus, the rule to represent the arithmetic sequence is: \[ A(n) = 12n + 175 \]

Part 2: To find the eleventh bid (or the \( 11 \)-th term): \[ A(11) = 12(11) + 175 \] \[ A(11) = 132 + 175 \] \[ A(11) = 307 \]

Therefore, the eleventh bid is \( 307 \).

In summary:

  • The rule is \( A(n) = 12n + 175 \)
  • The eleventh bid is \( 307 \).