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

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 then^th term of this sequence, forn=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

GPT-4o mini · Answer

To find the $ n $th term of the arithmetic sequence given (5.30, 7.80, 10.30, 12.80...), we first need to identify the first term and the common difference. 1. **Identify the first term** ($ a_1 $): The first term ($ n = 1 $) is $ 5.30 $. 2. **Calculate the common difference** ($ d $): To find the common difference, we can subtract the first term from the second term: $$ d = 7.80 - 5.30 = 2.50 $$ The common difference is $ 2.50 $. 3. **General formula for an arithmetic sequence**: The formula for the $ n $th term of an arithmetic sequence is given by: $$ a_n = a_1 + (n-1) \cdot d $$ Plugging in the values for $ a_1 $ and $ d $: $$ a_n = 5.30 + (n-1) \cdot 2.50 $$ Simplifying this: $$ a_n = 5.30 + 2.50n - 2.50 $$ $$ a_n = 2.80 + 2.50n $$ Thus, the function that represents the $ n $th term of this sequence is: $$ f(n) = 2.80 + 2.50n $$ The correct choice from the options provided is: **D. $ f(n) = 2.80 + 2.50n $**.