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

Math,

Your recursive formula means that t_n-1 is the fare charged for n-1, or one less than n, kilometers.

Given t₁ = $8, we can find t₂: t₂ = t₁ + 0.75 = 8+.75 = $8.75. Likewise, t₃ = 8.75 + .75 = $9.50.

Our pattern will look like 8, 8.75, 9.5,... adding .75 each time.

If we start with 8 for our initial value, we'll get the formula .75n+8, which gives us t₁=8.75, but that's too much!

t₁ = 8. Start by using

t_n = .75(n-1) +8. Then the 0.75 isn't added until the n=2.

Distribute the .75 to get a simpler, explicit formula: t_n = .75(n-1) +8 => t_n = .75n - .75 +8 =>

t_n = .75n +7.25.

Try some values we got before to make sure the equation works, then plug in 15 to get the dost for 15 km:

t₁₅ = .75(15) + 7.25 = $18.50

This series is arithmetic because it has a constant rate of change (slope). Each additional kilometer increases the price by .75.

I hope this helps and answers all your questions!

Have fun mathing, and let me know if you need any more help.

LizZ