Problem 97 - Precalculus with Limits

Question

A county fair is holding a baked goods competition in which the top eight bakers receive cash prizes. First place receives a cash prize of \( \$200 \),second place receives \( \$175 \), third place receives \( \$150 \),and so on. (a) Write a sequence that represents the cash prize awarded in terms of the place in which the baked good places. (b) Find the total amount of prize money awarded at the competition.

Step-by-Step Solution

Answer

The sequence representing the cash prize awarded in terms of the finishing place of the baked good is given by \(a_n = 225 - 25n\). The total prize money awarded at the competition is \$900.

Step 1: Creating the Sequence

According to the details provided in the exercise, the first place receives \$200, the second place \$175, the third place \$150, and so on, with a difference of \$-25 between each consecutive placement. Therefore, a sequence is created where \(a_n = 225 - 25n\) where \(n\) is the place in which the baked good finished.

Step 2: Finding the Sum of the Sequence

In order to find the total prize money, we find the sum of this arithmetic sequence. The formula to find the sum of an arithmetic sequence is given by \[S_n = \frac{n}{2} (a_1 + a_n)\]Here, \(n = 8\) (as there are 8 prizes), \(a_1 = 200\) (prize for the first place), and \(a_n = a_8 = 225 - 25*8 = \$25\) (prize for the eighth place).Substitute these values into the formula, we get \[S_8 = \frac{8}{2} (200 + 25) = 4 * 225 = \$900\]

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence is a central concept when dealing with problems involving a series of numbers with a common difference. An arithmetic sequence is characterized by each term being equal to the previous term plus a constant difference. To find the sum of such a sequence, we use the formula:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Here:

• \(S_n\) is the sum of the first \(n\) terms of the sequence.
• \(n\) is the number of terms.
• \(a_1\) is the first term.
• \(a_n\) is the last term.

Plug the values from any specific problem you have into this formula to calculate the total of the series. For example, in a prize sequence where the rewards decrease by \\(25 per rank and begin at \\)200, knowing the number of ranks helps to compute the total prize money quickly.

Prize Sequence

In competitive events, a prize sequence may often follow an arithmetic pattern, especially when the prizes become smaller in predictable intervals. Understanding how to construct such a sequence is crucial to solving related problems. In our example, the county fair awards cash prizes starting at \\(200 for first place, \\)175 for second place, \\(150 for third, and so on. The difference between these consecutive amounts is \\)-25. Thus, you form a sequence that follows this pattern:

• \(a_1 = \\(200\) for first place
• \(a_2 = \\)175\) for second place
• and so forth until \(a_8 = \$25\)

This sequence can be represented by the formula \(a_n = 225 - 25n\), where \(n\) represents the ranking position. By using such a formula, calculating any prize amount becomes easy.

Arithmetic Series Formula

The arithmetic series formula is a crucial tool in finding the sum of sequences where each term has a constant difference from the next. The formula \[ S_n = \frac{n}{2} (a_1 + a_n) \]allows for the efficient computation of the sum of terms in an arithmetic sequence. The arithmetic sequence must adhere to the form where the difference between any two consecutive terms is constant, like the decreasing prizes at our county fair, where each lower prize is \$25 less than the previous.Understanding how to use and apply this formula is crucial in determining total amounts, like the total prizes awarded in our example. It involves identifying the key values:\(a_1\) as the starting term, \(a_n\) as the ending term, and \(n\) as the total number of terms. Each of these inputs helps provide a simple yet effective way to sum a series without unnecessarily lengthy calculations.