Question

Type the correct answer in the box

A new clothing store uses advertising through social media to draw customers and increase the number of transactions each month. The

number of transactions, n), for n months is shown in the table below.

Month, n No. of Transactions, f(n)

1

10

2

20

3

40

4

80

If the number of transactions per month continues to increase in this way, fill in the values for a and rto create the function that

describes this situation.

Answer 1

The function that describes the situation is
`\( f(n) = 10 * 2^((n-1)) \)`.

Step-by-step explanation:

The given table represents a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio. In this case, the first term (a) is 10, and the common ratio (r) is 2 because each transaction count is doubled each month.

The general formula for a geometric sequence is
`\(f(n) = a * r^((n-1))\)`, where f(n) is the term at position n, a is the first term, r is the common ratio, and n is the position of the term.

By substituting the given values, we get
`\(f(n) = 10 * 2^((n-1))\)`, which represents the number of transactions each month based on the given pattern.

Answer 2

The function that describes the number of transactions
`(f(n))` for the given situation is
`\( f(n) = 10 * 2^(n-1) \)`.

Step-by-step explanation:

In the given table, the number of transactions doubles each month. This indicates a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio. The initial term (a) can be identified as 10, and the common ratio (r) is 2 since each term is twice the preceding one.

The general form of a geometric sequence is
`\( f(n) = a * r^((n-1)) \)`. Substituting the given values, we get
`\( f(n) = 10 * 2^(n-1) \)`. Here, ( n ) represents the number of months.

This formula makes sense because, in each month, the number of transactions is twice the number of transactions in the previous month. The term
`\( 2^(n-1) \)` accounts for this doubling effect, and by starting with 10 transactions in the first month, the formula accurately models the entire sequence.