The following recurrence relation can generate a sequence of numbers.
`T_0 = 10, qquad T_(n + 1) = T_n + 3`
The number 13 appears in this sequence as
- `T_1`
- `T_2`
- `T_3`
- `T_10`
- `T_13`
Recursion, GEN1 2019 NHT 17 MC
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 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 8 MC
Paula started a stamp collection. She decided to buy a number of new stamps every week.
The number of stamps bought in the `n`th week, `t_n`, is defined by the difference equation
`t_n = t_(n-1) + t_(n-2)\ \ \ text(where)\ \ \ t_1 = 1 and t_2 = 2`
The total number of stamps in her collection after five weeks is
A. `8`
B. `12`
C. `15`
D. `19`
E. `24`
CORE*, FUR1 2006 VCAA 5 MC
A difference equation is defined by
`f_(n+1) - f_n = 5\ \ \ \ \ text (where)\ \ f_1 =– 1`
The sequence `f_1, \ f_2, \ f_3, ...` is
A. `5, 4, 3\ …`
B. `4, 9, 14\ …`
C. `– 1, – 6, – 11\ …`
D. `– 1, 4, 9\ …`
E. `– 1, 6, 11\ …`
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 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 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 5 MC
The difference equation
`t_(n+2) = t_(n+1) + t_n` where `t_1 = a` and `t_2 = 7`
generates a sequence with `t_5 = 27`.
The value of `a` is
A. 0
B. 1
C. 2
D. 3
E. 4
CORE*, FUR1 2014 VCAA 7 MC
The first term of a Fibonacci-related sequence is `p`.
The second term of the same Fibonacci-related sequence is `q`.
The difference in value between the fourth and fifth terms of this sequence is
A. `p - q`
B. `q - p`
C. `p + q`
D. `p + 2q`
E. `2p + 3q`
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`
CORE*, FUR1 2012 VCAA 1 MC
1, 9, 10, 19, 29, . . .
The sixth term of the Fibonacci-related sequence shown above is
A. 30
B. 39
C. 40
D. 48
E. 49
CORE*, FUR1 2009 VCAA 1 MC
The first six terms of a Fibonacci related sequence are shown below.
4, 7, 11, 18, 29, 47, ...
The next term in the sequence is
A. 58
B. 65
C. 76
D. 94
E. 123
CORE*, FUR1 2013 VCAA 5 MC
A sequence is generated by the difference equation
`t_(n+1)=2 xx t_n,\ \ \ \ \ t_1=1`
The `n`th term of this sequence is
A. `t_n=1×2^(n-1)`
B. `t_n=1+2^(n-1)`
C. `t_n=1+2×(n-1)`
D. `t_n=2+(n-1)`
E. `t_n=2+1^(n-1)`