In one city, a taxi fare includes an initial fixed charge, plus a charge based on each kilometre travelled. Determine the following...

Question

In one city, a taxi fare includes an initial fixed charge, plus a charge based on each kilometre travelled. The possible fares charged, in dollars, in this city, can be represented by the recursion formula

t_n = t_n-1 + 0.75;  t₁ =8

a) What is the initial fixed charge?

b) What is the charge per kilometre travelled?

c) Is this an arithmetic or geometric sequence?

d) Find the general term of the sequence.

e) Calculate the fare for a 15 km trip.

Please answer in an organized and neat/easy to read format. And please only include the information that would be understood by a Grade 11. Thank you :)

Answer 1

a) Assuming that t₁ represents the charge after n=1 kilometers of travel, the initial charge, t₀, would be $7.25 and can be calculated by plugging n=1 into the recursion formula:

t₁ = t₀ + 0.75

8 = t₀ + 0.75

-0.75 -0.75

7.25 = t₀

b) According to the recursion formula each number in the sequence is 0.75 more than the previous. This means that each additional mile costs $0.75.

c) This is an arithmetic sequence since each new term is being produced by adding 0.75 to the previous term.

d) The "general term" or general formula of this sequence means that they want the explicit form of the arithmetic sequence. The explicit form will allow us to simply plug in a number of kilometers and get the fare. For that we simply need to plug the first term (t₀ = 7.25) and common difference (d = 0.75) into the following:

t(n) = t₀ + (n)d

t(n) = 7.25 + (n)(0.75)

So the "general term" or formula is t(n) = 7.25 + (n)(0.75)

e) To find the cost of a 15 km trip you simply need to plug n=15 into the explicit or "general formula":

t(15) = 7.25 + (15)(0.75)

t(15) = 7.25 + 11.25

t(15) = 18.5