A sequence of numbers is generated by the recurrence relation shown below.
`P_0 = 2,quadqquadP_(n + 1) = 3P_n - 1`
What is the value of `P_3`?
- 2
- 5
- 11
- 41
- 122
CORE, FUR1 2019 VCAA 17 MC
Consider the recurrence relation shown below.
`A_0 = 3, qquad A_(n + 1) = 2A_n + 4`
The value of `A_3` in the sequence generated by this recurrence relation is given by
- `2 xx 3 + 4`
- `2 xx 4 + 4`
- `2 xx 10 + 4`
- `2 xx 24 + 4`
- `2 xx 52 + 4`
CORE, FUR1 2017 VCAA 18 MC
The first five terms of a sequence are 2, 6, 22, 86, 342 …
The recurrence relation that generates this sequence could be
- `P_0 = 2,qquadP_(n + 1) = P_n + 4`
- `P_0 = 2,qquadP_(n + 1) = 2 P_n + 2`
- `P_0 = 2,qquadP_(n + 1) = 3 P_n`
- `P_0 = 2,qquadP_(n + 1) = 4 P_n – 2`
- `P_0 = 2,qquadP_(n + 1) = 5 P_n – 4`
CORE, FUR1 2016 VCAA 17 MC
Consider the recurrence relation below.
`A_0 = 2,\ \ \ \ \ A_(n + 1) = 3 A_n + 1`
The first four terms of this recurrence relation are
- `0, 2, 7, 22\ …`
- `1, 2, 7, 22\ …`
- `2, 5, 16, 49\ …`
- `2, 7, 18, 54\ …`
- `2, 7, 22, 67\ …`
CORE*, FUR1 2015 VCAA 9 MC
Paul has to replace 3000 m of fencing on his farm.
Let `F_n` be the length, in metres, of fencing left to replace after `n` weeks.
The difference equation
`F_(n + 1) = 0.95F_n + a\ \ \ \ \ \ F_0 = 3000`
can be used to calculate the length of fencing left to replace after `n` weeks.
In this equation, `a` is a constant.
After one week, Paul still has 2540 m of fencing left to replace.
After three weeks, the length of fencing, in metres, left to replace will be closest to
A. 1310
B. 1380
C. 1620
D. 1690
E. 2100
CORE*, FUR1 2015 VCAA 6 MC
Miki is competing as a runner in a half-marathon.
After 30 minutes, his progress in the race is modelled by the difference equation
`K_(n + 1) = 0.99K_n + 250,\ \ \ \ \ \ K_30 = 7550`
where `n ≥ 30` and `K_n` is the total distance Miki has run, in metres, after `n` minutes.
Using this difference equation, the total distance, in metres, that Miki is expected to have run 32 minutes after the start of the race is closest to
A. 7650
B. 7725
C. 7800
D. 7900
E. 8050
CORE*, FUR1 2006 VCAA 3-4 MC
The following information relates to Parts 1 and 2.
A farmer plans to breed sheep to sell.
In the first year she starts with 50 breeding sheep.
During the first year, the sheep numbers increase by 84%.
At the end of the first year, the farmer sells 40 sheep.
Part 1
How many sheep does she have at the start of the second year?
A. 2
B. 42
C. 52
D. 84
E. 92
Part 2
If `S_n` is the number of sheep at the start of year `n`, a difference equation that can be used to model the growth in sheep numbers over time is
| A. `S_(n+1) = 1.84S_n - 40` | `\ \ \ \ \ text(where)\ \ S_1 = 50` | |
| B. `S_(n+1) = 0.84S_n - 50` | `\ \ \ \ \ text(where)\ \ S_1 = 40` | |
| C. `S_(n+1) = 0.84S_n - 40` | `\ \ \ \ \ text(where)\ \ S_1 = 50` | |
| D. `S_(n+1) = 0.16S_n - 50` | `\ \ \ \ \ text(where)\ \ S_1 = 40` | |
| E. `S_(n+1) = 0.16S_n - 40` | `\ \ \ \ \ text(where)\ \ S_1 = 50` |
CORE*, FUR1 2007 VCAA 3 MC
The difference equation
`t_(n+1) = at_n + 6 quad text (where) quad t_1 = 5`
generates the sequence
`5, 21, 69, 213\ …`
The value of `a` is
A. – 1
B. 3
C. 4
D. 15
E. 16
CORE*, FUR1 2013 VCAA 8 MC
The initial rate of pay for a job is $10 per hour.
A worker’s skill increases the longer she works on this job. As a result, the hourly rate of pay increases each month.
The hourly rate of pay in the `n`th month of working on this job is given by the difference equation
`S_(n+1) = 0.2 xx S_n+15\ \ \ \ \ \ S_1 = 10`
The maximum hourly rate of pay that the worker can earn in this job is closest to
A. $3.00
B. $12.00
C. $12.50
D. $18.75
E. $75.00
CORE*, FUR1 2009 VCAA 8 MC
A patient takes 15 milligrams of a prescribed drug at the start of each day.
Over the next 24 hours, 85% of the drug in his body is used. The remaining 15% stays in his body.
Let `D_n` be the number of milligrams of the drug in the patient’s body immediately after taking the drug at the start of the `n`th day.
A difference equation for determining `D_(n+1)`, the number of milligrams in the patient’s body immediately after taking the drug at the start of the `n+1`th day, is given by
| A. `D_(n + 1) = 85 D_n + 15` | `D_1 = 15` |
| B. `D_(n + 1) = 0.85 D_n + 15` | `D_1 = 15` |
| C. `D_(n + 1)= 0.15 D_n + 15` | `D_1 = 15` |
| D. `D_(n + 1)= 0.15 D_n + 0.85` | `D_1 = 15` |
| E. `D_(n + 1)= 15 D_n + 85` | `D_1 = 15` |
CORE*, FUR1 2009 VCAA 7 MC
The difference equation `u_(n + 1) = 4u_n - 2` generates a sequence.
If `u_2 = 2`, then `u_4` will be equal to
A. 4
B. 8
C. 22
D. 40
E. 42
CORE*, FUR1 2011 VCAA 7 MC
Let `P_2011` be the number of pairs of shoes that Sienna owns at the end of 2011.
At the beginning of 2012, Sienna plans to throw out the oldest 10% of pairs of shoes that she owned in 2011.
During 2012 she plans to buy 15 new pairs of shoes to add to her collection.
Let `P_2012` be the number of pairs of shoes that Sienna owns at the end of 2012.
A rule that enables `P_2012` to be determined from `P_2011` is
A. `P_2012 = 1.1 P_2011 + 15`
B. `P_2012 = 1.1 (P_2011 + 15)`
C. `P_2012 = 0.1 P_2011 + 15`
D. `P_2012 = 0.9 (P_2011 + 15)`
E. `P_2012 = 0.9 P_2011 + 15`
CORE*, FUR1 2014 VCAA 9 MC
Sam takes a tablet containing 200 mg of medicine once every 24 hours.
Every 24 hours, 40% of the medicine leaves her body. The remaining 60% of the medicine stays in her body.
Let `D_n` be the number of milligrams of the medicine in Sam’s body immediately after she takes the `n`th tablet.
The difference equation that can be used to determine the number of milligrams of the medicine in Sam’s body immediately after she takes each tablet is shown below.
`D_(n + 1) = 0.60D_n + 200,\ \ \ \ \ \ D_1 = 200`
Which one of the following statements is not true?
A. The number of milligrams of the medicine in Sam’s body never exceeds 500.
B. Immediately after taking the third tablet, 392 mg of the medicine is in Sam’s body.
C. The number of milligrams of the medicine that leaves Sam’s body during any 24-hour period will always be less than 200.
D. The number of milligrams of the medicine that leaves Sam’s body during any 24-hour period is constant.
E. If Sam stopped taking the medicine after the fifth tablet, the amount of the medicine in her body would drop to below 200 mg after a further 48 hours.
CORE*, FUR1 2012 VCAA 9 MC
Three years after observations began, 12 300 birds were living in a wetland.
The number of birds living in the wetland changes from year to year according to the difference equation
`t_(n+ 1) = 1.5t_n - 3000, quad quad t_3 = text (12 300)`
where `t_n` is the number of birds observed in the wetland `n` years after observations began.
The number of birds living in the wetland one year after observations began was closest to
A. `8800`
B. `9300`
C. `10\ 200`
D. `12\ 300`
E. `120\ 175`