Question

The arithmetic sequence below shows the costs, in dollars, of riding 1, 2, 3, and 4 miles (and so on) in a certain taxicab.

5.30, 7.80, 10.30, 12.80, ...

Which function represents the n^th
term of this sequence, for n= 1, 2, 3, 4, ...
?

Answer 1

To find the \( n \)th term of the given arithmetic sequence, we first need to identify the first term and the common difference.

The sequence given is:
- For \( n = 1 \): 5.30
- For \( n = 2 \): 7.80
- For \( n = 3 \): 10.30
- For \( n = 4 \): 12.80

### Step 1: Identify the first term (\( a \))
The first term \( a = 5.30 \).

### Step 2: Find the common difference (\( d \))
We can find the common difference by subtracting any term from the subsequent term:
- \( 7.80 - 5.30 = 2.50 \)
- \( 10.30 - 7.80 = 2.50 \)
- \( 12.80 - 10.30 = 2.50 \)

So, the common difference \( d = 2.50 \).

### Step 3: Write the general formula for the \( n \)th term of an arithmetic sequence
The formula for the \( n \)th term of an arithmetic sequence is given by:
\[
a_n = a + (n - 1) \cdot d
\]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 4: Substitute the values
Substituting \( a = 5.30 \) and \( d = 2.50 \):
\[
a_n = 5.30 + (n - 1) \cdot 2.50
\]

### Step 5: Simplify the equation
Distributing the \( 2.50 \):
\[
a_n = 5.30 + 2.50n - 2.50
\]
\[
a_n = 2.50n + 2.80
\]

### Conclusion
The function that represents the \( n \)th term of this sequence is:
\[
a_n = 2.50n + 2.80
\]