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 identify the first term (\( a_1 \)) and the common difference (\( d \)).

From the sequence:
- \( a_1 = 5.30 \) (cost for 1 mile)
- \( a_2 = 7.80 \) (cost for 2 miles)
- \( a_3 = 10.30 \) (cost for 3 miles)
- \( a_4 = 12.80 \) (cost for 4 miles)

Next, we can calculate the common difference \( d \):
\[
d = a_2 - a_1 = 7.80 - 5.30 = 2.50
\]
\[
d = a_3 - a_2 = 10.30 - 7.80 = 2.50
\]
\[
d = a_4 - a_3 = 12.80 - 10.30 = 2.50
\]

The common difference is consistent at \( d = 2.50 \).

The formula for the \( n^{th} \) term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1) \cdot d
\]

Substituting the values of \( a_1 \) and \( d \):
\[
a_n = 5.30 + (n - 1) \cdot 2.50
\]

Now simplifying this:
\[
a_n = 5.30 + 2.50n - 2.50
\]
\[
a_n = 2.50n + 2.80
\]

Thus, the function that represents the \( n^{th} \) term of this sequence is:
\[
a_n = 2.50n + 2.80
\]

for \( n = 1, 2, 3, 4, \ldots \).