Number patterns - difficult question!

[Wabba Fig!]

Over the last 60 years in Australia, the numbers of cars on the road has increased but proportionally the number of auto fatalities has decreased. The number of fatalities per 10,000 registered cars is given by:
t_n+1 = 0.9975t_n - 0.09     ,      t₁ = 16  for n= 1, 2, 3 .......where year 1 is 1945

a) What would the fatality rate per 10,000 cars, be in 2000?
b) In 2005 there were in fact 1,636 fatalities in Australia. How many registered vehicles were there in Australia in 2005?


I know how to do a) already, this is what I did:
a) 2000-1945 = 55
     55+1 = 56

But I have no idea how to do b)

please help!

[Zealous]

Hi, when you have to use a difference equation to predict data that is very far away (from 1945 to 200 is 55 years.) it is very useful to set up a table in your calculator.

I use the CAS CX so I'll let you know the instructions for that, but you may have a different calculator so I can't help you.

1) Open Lists and Spreadsheets.
2) Use the arrow keys to select the 2nd row down of ANY column, so it should be grey. (So if you just opened list and spreadsheets, it should be selected on A1, you want to go up above A1 to the row with the diamond, which is grey.)
3) On the keypad: Menu -> Data -> Generate Sequence.
4) For this example use:

- This is the difference equation.

- the very first term of the sequence.

5) You can pretty much leave the rest in the window as is and click generate.

Now that you have generated your sequence, you can scroll down through the list until you get to the 56th row.

Because they are saying that in 1945 there were 16 fatalities per 10,000. That's why you are looking for the 56th term as 1945 is the first term.

My spreadsheet showed me: 9.31198 (fatalities per 10,000)
(can you check your answers if you have them to make sure I've come up with the right answer?)

Part B:
If you scroll down even further in your generated list, to Row 61, you will find the amount of fatalities per 10,000 cars in 2005 is 8.74841.

So we know a few bits of information. There were 8.748 fatalities per 10,000 registered cars. And in 2005 there were 1,646 fatalities in Australia.

My method is as bit awkward, in a way, but what I did was:



I found the amount of times the fatalities per 10,000 goes into the 1646 fatalities in 2005, then multiplied it by 10,000 [cars]. If someone can explain it better than that, go for it!