Question

Catherine found that as she increases the price of a chocolate bar, the number of sales per week decreased.a. Write a formula for the arithmetic sequence that represents the number of sales per week.b. Explain how you could use the information to predict the number of sales for the eighth week.

Answer 1

Given:

The table shows an arithmetic sequence.

Part A:

The nth term of an arithmetic progression is given by:

`\begin{gathered} a_n=a_1+(n-1)d \\ \\ where: \\ a_1\text{ is the first term} \\ n\text{ is the number of weeks} \\ d\text{ is the common difference} \end{gathered}`

Hence,

`\begin{gathered} a_1=149 \\ d=143-149=-6 \\ Hence,\text{ the formula is:} \\ a_n=149+(n-1)(-6) \\ a_n=149-6n+6 \\ a_n=149+6-6n \\ a_n=155-6n \end{gathered}`

Therefore, the formula for the arithmetic sequence that represents the number of sales per week is;

`a_n=155-6n`

Part B:

The number of sales in the eighth week is:

`\begin{gathered} when\text{ n = 8} \\ a_n=155-6n \\ a_8=155-6(8) \\ a_8=155-48 \\ a_8=107 \end{gathered}`

Therefore, the number of sales expected for the eighth week is 107.