Let

A (n)

be the arithmetic sequence representing the amount of money left on the card. At the beginning, the cafeteria card's value is

$ 50 .

This means that our first term,

A (1),

is

50 .

After the first purchase on Monday, its value is

$ 46.75,

and after the next day its value is

$ 43.50 .

50 ⟶ - 3.25 46.75 ⟶ - 3.25 43.5 ⟶ - 3.25 \dots

As we can see, the common difference

d

is

- 3.25 .

Let's recall the form of an explicit formula of an arithmetic sequence.

A (n) = A (1) + (n - 1) d

In the above formula,

A (1)

is the initial value and

d

is the common difference. We will get our formula by substituting

A (1) = 50

and

d = - 3.25 .

A (n) = A (1) + (n - 1) d

$A (1) = 50$ , $d = - 3.25$

A (n) = 50 + (n - 1) (- 3.25)

Commutative Property of Multiplication

A (n) = 50 + (- 3.25) (n - 1)

$a + (- b) = a - b$

A (n) = 50 - 3.25 (n - 1)

This formula gives us the terms of the arithmetic sequence formed by the amount of money left on the card.

Let's review our sequence again.

50, 46.75, 43.5, \dots

Notice that the

2^{nd} term

of the sequence is the amount of money left on the card after buying

1

lunch, the

3^{rd} term

is the amount of money left on the card after buying

2

lunches, and so on. Therefore, the

1 3^{th} term

of the sequence will give us the amount of money left on the card after buying

12

lunches.

A (n) = 50 - 3.25 (n - 1)

$n = 13$

A (13) = 50 - 3.25 (13 - 1)

▼

Evaluate right-hand side

Subtract term

A (13) = 50 - 3.25 (12)

A (13) = 50 - 39

Subtract term

A (13) = 11

This means that after buying

12

lunches, the card's value will be

$ 11 .

Exercise