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, ... ?

A) f(n) = -2.50 + 7.80n
B) f(n) = 5.30 + 7.80n
C) f(n) = 5.30 + 2.50n
D) f(n) = 2.80 + 2.50n

Answer 1

To determine the function that represents the \(n\)th term of the given arithmetic sequence, let's first identify the initial term (first term) and the common difference.

The sequence provided is:
- For \(n = 1\), the cost is \(5.30\)
- For \(n = 2\), the cost is \(7.80\)
- For \(n = 3\), the cost is \(10.30\)
- For \(n = 4\), the cost is \(12.80\)

1. **Initial term (\(a\))**: The first term \(a = 5.30\).

2. **Common difference (\(d\))**: We calculate the difference between consecutive terms:
- \(7.80 - 5.30 = 2.50\)
- \(10.30 - 7.80 = 2.50\)
- \(12.80 - 10.30 = 2.50\)

The common difference \(d = 2.50\).

Given that this is an arithmetic sequence, the \(n\)th term can be represented by the formula:
\[
f(n) = a + (n - 1)d
\]
Here, substituting in the values we have:
\[
f(n) = 5.30 + (n - 1) \cdot 2.50
\]
Simplifying this gives:
\[
f(n) = 5.30 + 2.50n - 2.50
\]
\[
f(n) = (5.30 - 2.50) + 2.50n
\]
\[
f(n) = 2.80 + 2.50n
\]

Therefore, the function that represents the \(n\)th term of the sequence is:

**D) \(f(n) = 2.80 + 2.50n\)**.