Chapter 9.2, Problem 74E

a.

To determine

To find the nth term (a_n) of a sequence that represents the cash prize received in terms of the place n the baked good is awarded.

Answer to Problem 74E

  $\begin{array}{l}{a}_n=225-25n\\ n=1,2,3,\mathrm{...},8\end{array}$

Explanation of Solution

Given:

In a baked goods competition,

First place receives $200

Second place receives $175,

Third place receives $150, and so on

Formula used:

The nth term of an arithmetic sequence is given by the formula

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ and d are first term and common ratio respectively.

Calculation:

First place receives $200, second place receives $175, and third place receives $150, and so on..

This can be written as a sequence, $200,175,150,\mathrm{.....}$ up to 8 terms, which is an arithmetic sequence with first term as $a_1=200$ and common difference as $d=200-175=-25$ .

The nth term of an arithmetic sequence is given by the formula

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ and d are first term and common ratio respectively.

Therefore, the nth term (a_n) of a sequence that represents the cash prize received is

  $\begin{array}{l}{a}_n=200+\left(n-1\right)\times \left(-25\right)\\ a_n=200-25n+25\\ a_n=225-25n\end{array}$

Where $n=1,2,3,\mathrm{...},8$

Conclusion:

Therefore, the nth term is $a_n=225-25n.$

b.

To determine

To find the total amount of prize money awarded at the competition.

Answer to Problem 74E

$900

Explanation of Solution

Given:

In a baked goods competition,

First place receives $200

Second place receives $175,

Third place receives $150, and so on

Formula used:

The sum of n terms of an arithmetic sequence is given by the formula,

  $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$

Where $a_1$ and d are first term and common ratio respectively.

Calculation:

The total amount of prize money awarded at the competition is the sum of 8 terms of an arithmetic sequence whose first term is $a_1=200$ and common difference is $d=-25$ is equal to

  $\begin{array}{l}\Rightarrow S_8=\frac{8}{2}\left[2\times 200+\left(8-1\right)\left(-25\right)\right]\\ \Rightarrow S_8=\frac{8}{2}\left[400+\left(7\right)\left(-25\right)\right]\\ \Rightarrow S_8=4\left[400-175\right]\\ \Rightarrow S_8=\$900\end{array}$

Conclusion:

Therefore, the final amount of prize money awarded is $900.