APs and GPs

CORE*, FUR1 2015 VCAA 8 MC

`A_n` is the `n`th term in a sequence.

Which one of the following expressions does not define a geometric sequence?

A. `A_(n + 1) = n`	`\ \ \ \ A_0 = 1`
B. `A_(n + 1) = 4`	`\ \ \ \ A_0 = 4`
C. `A_(n + 1) = A_n + A_n`	`\ \ \ \ A_0 = 3`
D. `A_(n + 1) = –A_n`	`\ \ \ \ A_0 = 5`
E. `A_(n + 1) = 4A_n`	`\ \ \ \ A_0 = 2`

Show Answers Only

`A`

Show Worked Solution

`text(Test all options by looking at the first)`

♦♦ Mean mark 22%.
MARKERS’ COMMENT: A routine way to solve this type of question is to write out the first few terms of each sequence.)

`text(3 terms that each produces.)`

`text(Consider)\ A,`

`A_0=1, \ A_1=0, \ A_2=1`

`text(There is no common ratio in this sequence.)`

`text(All other options can be shown to have a common ratio.)`

`=> A`

PATTERNS, FUR 1 2006 VCAA 9 MC

A healthy eating and gym program is designed to help football recruits build body weight over an extended period of time.

Roh, a new recruit who initially weighs 73.4 kg, decides to follow the program.

In the first week he gains 400 g in body weight.

In the second week he gains 380 g in body weight.

In the third week he gains 361 g in body weight.

If Roh continues to follow this program indefinitely, and this pattern of weight gain remains the same, his eventual body weight will be closest to

A. `74.5\ text(kg)`

B. `77.1\ text(kg)`

C. `77.3\ text(kg)`

D. `80.0\ text(kg)`

E. `81.4\ text(kg)`

Show Answers Only

`E`

Show Worked Solution

`text(Checking for a common ratio,)`

♦♦ Mean mark 33%.

`t_2/t_1`	`=380/400=0.95`
`t_3/t_2`	`=361/380=0.95`

`:.\ text(GP where)\ \ \ a`	`=400, and`
`r`	`=0.95`

`text(S)text(ince)\ \ |\ r\ |<1,`

`S_oo`	`=a/(1-r)`
	`=400/(1-0.95)`
	`=8000\ text(g)`

`:.\ text(Net weight gain will be 8 kg which will make Roh’s)`

`text{eventual weight 73.4 + 8 = 81.4 kg.}`

`rArr E`

PATTERNS, FUR1 2006 VCAA 6 MC

A crystal measured 12.0 cm in length at the beginning of a chemistry experiment.

Each day it increased in length by 3%.

The length of the crystal after 14 days growth is closest to

A. `12.4\ text(cm)`

B. `16.7\ text(cm)`

C. `17.0\ text(cm)`

D. `17.6\ text(cm)`

E. `18.2\ text(cm)`

Show Answers Only

`E`

Show Worked Solution

`text(After day 1, length)\ = 12.0 xx 1.03`

♦♦ Mean mark 25%.
MARKERS’ COMMENT: The question asks for length after 14 days’ growth, not its length on the start of the 14th day.

`text(After day 2, length)\ = 12.0 xx 1.03^2`

`vdots`

`:.\ text(After day 14, length)`

`= 12.0 xx 1.03^14 `

`= 18.15\ text(cm)`

`rArr E`

PATTERNS, FUR1 2006 VCAA 1 MC

Which one of the following sequences shows the first five terms of an arithmetic sequence?

A. `1, 3, 9, 27, 81\ …`

B. `1, 3, 7, 15, 31\ …`

C. `– 10, – 5, 5, 10, 15\ …`

D. `– 4, – 1, 2, 5, 8\ …`

E. `1, 3, 8, 15, 24\ …`

Show Answers Only

`D`

Show Worked Solution

`text(Only D has a consistent common difference)`

`text(between all values.)`

`rArr D`

PATTERNS, FUR1 2007 VCAA 7 MC

The first term, `t_1`, of a geometric sequence is positive.

The common ratio of this sequence is negative.

A graph that could represent the first five terms of this sequence is

Show Answers Only

`E`

Show Worked Solution

`text(General terms are)\ \ \ a, ar, ar^2, …`

`text(By elimination,)`

`a >0,\ text(Eliminate D.)`

`r<0,\ \ text(terms will alternate between positive and)`

`text(negative. Eliminate A, B and C.`

`rArr E`

PATTERNS, FUR1 2007 VCAA 1 MC

For the geometric sequence

`24, 6, 1.5\ …`

the common ratio of the sequence is

A. `– 18`

B. `0.25`

C. `0.5`

D. `4`

E. `18`

Show Answers Only

`B`

Show Worked Solution

`r=t_2/t_1=6/24=0.25`

`rArr B`

PATTERNS, FUR1 2008 VCAA 9 MC

When full, a swimming pool holds 50 000 litres of water.

Due to evaporation and spillage the pool loses, on average, 2% of the water it contains each week.

To help to make up this loss, 500 litres of water is added to the pool at the end of each week.

Assume the pool is full at the start of Week 1.

At the start of Week 5 the amount of water (in litres) that the pool contains will be closest to

A. `47\ 500`

B. `47\ 600`

C. `48\ 000`

D. `48\ 060`

E. `48\ 530`

Show Answers Only

`D`

Show Worked Solution

`text(Let)\ \ V_n=\ text(Volume at the start of week)\ n`

`V_1=50\ 000`

♦ Mean mark 41%.

`V_2=(50\ 000)0.98+500 =49\ 500`

`V_3=(49\ 500)0.98+500 =49\ 010`

`V_4=(49\ 010)0.98+500 =49\ 529.8`

`V_5=(48\ 529.8)0.98+500=48\ 059.2…`

`=>D`

CORE*, FUR1 2008 VCAA 8 MC

When placed in a pond, the length of a fish was 14.2 centimetres.

During its first month in the pond, the fish increased in length by 3.6 centimetres.

During its `n`th month in the pond, the fish increased in length by `G_n` centimetres, where `G_(n+1) = 0.75G_n`

The maximum length this fish can grow to (in cm) is closest to

A. 14.4

B. 16.9

C. 19.0

D. 28.6

E. 17.2

Show Answers Only

`D`

Show Worked Solution

`text(Initial length) = 14.2\ text(cm)`

♦ Mean mark 45%.

`G_1 = 3.6`

`G_2 = (0.75) G_1 = (0.75) 3.6`

`G_3 = (0.75) G_2 = (0.75^2) 3.6`

`text(Growth is a geometric sequence)`

`underbrace(3.6, \ 3.6(0.75), \ 3.6(0.75)^2, …)_{text(GP where)\ \ a=3.6,\ \ r=0.75}`

`text(S)text(ince)\ \ |\ r\ |< 1,`

`S_oo`	`= a/(1-r)`
	`= (3.6)/(1-0.75)`
	`= 14.4`

`:.\ text(Maximum length of fish)`

`= 14.2 +14.4`

`=28.6\ text(cm)`

`=> D`

CORE*, FUR1 2008 VCAA 7 MC

The sequence `12, 15, 27, 42, 69, 111 …` can best be described as

A. fibonacci-related

B. arithmetic with `d > 1`

C. arithmetic with `d < 1`

D. geometric with `r > 1`

E. geometric with `r < 1`

Show Answers Only

`A`

Show Worked Solution

`12, 15, 27, 42, 69, 111\ …`

`T_3 = T_1 + T_2`

`T_4 = T_3 + T_2`

`=> A`

PATTERNS, FUR1 2008 VCAA 5 MC

The first term of a geometric sequence is 9.

The third term of this sequence is 121.

The second term of this sequence could be

A. `– 65`

B. `– 33`

C. `56`

D. `65`

E. `112`

Show Answers Only

`B`

Show Worked Solution

`text(GP where) \ \ \ T_1=9, and T_3=121`

`T_1`	`=a=9`
`T_3`	`=ar^2=121`
`:.9r^2`	`=121`
`r^2`	`=121/9`
`r`	`= +-11/3`

`text(If)\ \ r=11/3,`

`T_2=ar=9 xx 11/3 = 33`

`text(If)\ \ r= – 11/3,`

`T_2=9 xx – 11/3 = – 33`

`=>B`

PATTERNS, FUR1 2008 VCAA 2 MC

For an examination, 8600 examination papers are to be printed at a rate of 25 papers per minute. After one hour, the number of examination papers that still need to be printed is

A. `1600`

B. `2500`

C. `6100`

D. `7100`

E. `8575`

Show Answers Only

`D`

Show Worked Solution

`text(Printing rate = 25 per minute)`

`:.\ text(In 1 hour, exam papers printed)`

`= 60xx25`

`= 1500`

`:.\ text(Papers still to be printed)`

`= 8600-1500`

`= 7100`

`=>D`

PATTERNS, FUR1 2013 VCAA 9 MC

Eleven speed bumps are placed on a road.

The speed bumps are placed so that the distance between consecutive speed bumps decreases according to an arithmetic sequence.

The distance between the first and last speed bumps is exactly 100 m.

The smallest distance between consecutive speed bumps is 2 m.

The largest distance, in m, between two consecutive speed bumps is

A. `16`

B. `18`

C. `20`

D. `22`

E. `24`

Show Answers Only

`B`

Show Worked Solution

`text(11 speed bumps over 100m distance has 10 intervals,)`

`text(so the series can be written)`

♦♦ Mean mark 33%.
MARKER’S COMMENT: The not often used formula `S_n=n/2(a+l)` provides the most efficient route to this solution, although `S_n=n/2[2a+(n-1)d]` can also be used after finding the value of `d`.

`a, a+d, a+2d, …, a+9d`

`text(Last distance)\ =a+9d=l=2\quad text{(given)}`

`S_10=100\ text(m)\ \ \ text{(given)}`

`text(Using)\ \ \ S_n`	`=n/2(a+l)`
`100`	`=10/2(a+2)`
`100`	`=5a+10`
`5a`	`=90`
`a`	`=18`

`:. 18text{m is largest distance (d is negative)}`

`=> B`

PATTERNS, FUR1 2013 VCAA 7 MC

The following are either three consecutive terms of an arithmetic sequence or three consecutive terms of a geometric sequence.

Which one of these sequences could not include 2 as a term?

A. `–1, 0.5, –0.25`

B. `–1, –3, –5`

C. `5, 12.5, 31.25`

D. `6, 8, 10`

E. `8, 16, 32`

Show Answers Only

`B`

Show Worked Solution

`text(By elimination,)`

`text(In A, GP where)\ \ r=0.5/-1=-0.5`

`:.\ text(Term before −1 is 2.)`

`text(In B, AP where)\ \ d=-2`

`:.\ text{Terms before −1 = −1 − (−2) = 1}`

`text{Term before 1 = 1 − (−2) = 3}`

`:.\ text(Does NOT include 2.)`

`text(Other choices can all be shown to contain 2.`

`=> B`

PATTERNS, FUR1 2009 VCAA 9 MC

There are 10 checkpoints in a 4500 metre orienteering course.

Checkpoint 1 is the start and checkpoint 10 is the finish.

The distance between successive checkpoints increases by 50 metres as each checkpoint is passed.

The distance, in metres, between checkpoint 2 and checkpoint 3 is

A. `225`

B. `275`

C. `300`

D. `350`

E. `400`

Show Answers Only

`D`

Show Worked Solution

`text(9 distances exist between 10 checkpoints)`

♦♦ Mean mark 26%.

`text(9 distances form an AP, where)`

`d = 50,\ \ \ S_9 = 4500`

`S_n`	`= n/2[2a + (n − 1)d]`
`4500`	`= 9/2 [2a + (9 − 1) × 50]`
`4500`	`= 9/2[2a + 400]`
`9a`	`= 2700`
`a`	`= 300`

`:.\ text(Distance between checkpoint 1 and 2)`

`= a=300\ text(m)`

`:.\ text(Distance between checkpoint 2 and 3)`

`= a+d=350\ text(m)`

`=> D`

PATTERNS, FUR1 2009 VCAA 5 MC

On Monday morning, Jim told six friends a secret. On Tuesday morning, those six friends each told the secret to six other friends who did not know it. The secret continued to spread in this way on Wednesday, Thursday and Friday mornings.

The total number of people (not counting Jim) who will know the secret on Friday afternoon is

A. `259`

B. `1296`

C. `1555`

D. `7776`

E. `9330`

Show Answers Only

`E`

Show Worked Solution

`text(Sequence is 6, 6 × 6, 6 × 6 × 6, …)`

♦♦ Mean mark 25%.
MARKERS’ COMMENT: Over 50% of students calculated the 5th term and not the sum of the first 5 terms!

`text(GP where)\ \ \ a`	`= 6, and`
`r`	`= 6 `

`S_n=`	`text(total number who know the secret)`
	`text(on the afternoon of day)\ n`

`text(On Friday,)\ \ n = 5`

`S_n`	`= (a (r^n − 1)) / (r −1)`
`:.S_5`	`= (6(6^5 −1)) / (6 −1)`
	`= 9330`

`=> E`

PATTERNS, FUR1 2009 VCAA 4 MC

The sum of the infinite geometric sequence 96, – 48, 24, –12, 6 . . . is equal to

A. `64`

B. `66`

C. `68`

D. `144`

E. `192`

Show Answers Only

`A`

Show Worked Solution

`text(Sequence is)\ \ \ 96, –48, 24, –12, 6, …`

`text(GP where)\ \ \ a`	`=96, and`
`r`	`=t_2/t_1 = (–48)/96 = –1 /2`

`text(S)text(ince)\ \ |\ r\ | < 1`

`S_∞`	`=a/(1-r)`
	`=96/(1-(-1 /2))`
	`=64`

`=> A`

PATTERNS, FUR1 2009 VCAA 3 MC

The first four terms of a geometric sequence are 6400, `t_2` , 8100, – 9112.5

The value of `t_2` is

A. `– 7250`

B. `– 7200`

C. `–1700`

D. `7200`

E. `7250`

Show Answers Only

`B`

Show Worked Solution

`text(GP is )6400, t_2, 8100, –9112.5`

`r`	`=t_2/t_1 = t_3/t_2`
`:. t_2 / 6400`	`= (–9112.5) / 8100`
`t_2`	`= (–9112.5 × 6400) / 8100`
	`= – 7200`

`=> B`

PATTERNS, FUR1 2009 VCAA 2 MC

Each week a young boy saves an amount of his pocket money. The amount saved forms part of an arithmetic sequence. The table shows the amounts he saves in weeks 1 to 3.

If he continues to save in this way, the amount he will save in week eight is

A. `$1.45`

B. `$1.60`

C. `$1.65`

D. `$7.40`

E. `$8.00`

Show Answers Only

`A`

Show Worked Solution

`text(Sequence is 0.40, 0.55, 0.70, …)`

`text(AP where)\ \ \ a`	`= 0.40`
`d`	`= 0.55 – 0.40 = 0.15`

`T_n`	`= a + (n – 1) \ d`
`:.T_8`	`= 0.40 + (8 – 1) \ 0.15`
	`= 0.40 + 1.05`
	`= 1.45`

`=> A`

PATTERNS, FUR1 2011 VCAA 9 MC

A toy train track consists of a number of pieces of track which join together.

The shortest piece of the track is 15 centimetres long and each piece of track after the shortest is 2 centimetres longer than the previous piece.

The total length of the complete track is 7.35 metres.

The length of the longest piece of track, in centimetres, is

A. `21`

B. `47`

C. `49`

D. `55`

E. `57`

Show Answers Only

`D`

Show Worked Solution

`text(Sequence is 15, 17, 19 , . . .)`

`text(AP where)\ \ \ a`	`= 15, and`
`d`	`= 2`

`S_n`	`= n / 2 [ 2a + (n – 1) d ] = 735\ text(cm)`
`:. 735`	`= n / 2 [ 2 xx 15 + (n – 1) 2 ]`
	`= n / 2 [ 30 + 2n – 2 ]`
	`= n^2 + 14n`
`0`	`= n^2 + 14n – 735`

`text(Using the quadratic formula)`

`n`	`= {–14 +- sqrt (14^2 – 4. 1. (–735))} / (2 xx 1)`
	`= (–14 +-56) / 2`
	`= 21 \ \ \ \ (n > 0)`

`:.\ text(Longest piece of track)`	`= a + (n – 1) d`
	`= 15 + (21 – 1) 2`
	`= 55\ text(cm)`

`=> D`

PATTERNS, FUR1 2011 VCAA 8 MC

The first three terms of an arithmetic sequence are `1, 3, 5 . . .`

The sum of the first `n` terms of this sequence, `S_n`, is

A. `S_n = n^2`

B. `S_n = n^2 - n`

C. `S_n = 2n`

D. `S_n = 2n - 1`

E. `S_n = 2n + 1`

Show Answers Only

`A`

Show Worked Solution

`text(Sequence is 1, 3, 5 , . . .)`

♦ Mean mark 38%.

`text(AP where)\ \ \ a`	`= 1, and`
`d`	`= 3 – 1 = 2`

`S_n`	`= n / 2 [ 2a + (n – 1) d ]`
	`= n / 2 [ 2 xx 1 + (n – 1) 2 ]`
	`= n / 2 [ 2 + 2n – 2 ]`
	`= n^2`

`=> A`

PATTERNS, FUR1 2011 VCAA 6 MC

For which one of the following geometric sequences is an infinite sum not able to be determined?

A. `4, 2, 1, 1 / 2\ . . .`

B. `1, 2, 4, 8\ . . .`

C. `–4, 2, –1, 1 / 2\ . . .`

D. `1, 1 / 2, 1 / 4, 1 / 8\ . . .`

E. `–1, 1/2, –1 / 4, 1 / 8\ . . .`

Show Answers Only

`B`

Show Worked Solution

`text(An infinite sum can only be determined when)`

`|\ r\ | <= 1`

♦ Mean mark 50%.

`text(In A,)\ \ \ r = t_2 / t_1 = 1 / 2`

`text(In B,)\ \ \ r = 2 / 1 = 2`

`text(In C,)\ \ \ r = 2 / (–4) = – 1 / 2`

`text(In D,)\ \ \ r = 1 / 2`

`text(In E,)\ \ \ r = – 1 / 2`

`=> B`

PATTERNS, FUR1 2011 VCAA 4 MC

The number of bees in a colony was recorded for three months and the results are displayed in the table below.

If this pattern of increase continues, which one of the following statements is not true.

A. There will be nine times as many bees in the colony in month 5 than in month 3.

B. In month 4, the number of bees will equal 270.

C. In month 6, the number of bees will equal 7290.

D. In month 8, the number of bees will exceed 20 000.

E. In month 10, the number of bees will be under 200 000.

Show Answers Only

`C`

Show Worked Solution

`text(Sequence is 10, 30, 90, …)`

`text(GP where)\ \ \ a`	` = 10, and`
`r`	` = t_2/t_1=30 / 10 = 3`

`text(In A,)\ \ T_5 = 10 xx 3^4 and T_3 = 10 xx 3^2`

`:. T_5 = T_3 xx 3^2\ \ text{(True)}`

`text(In B,)\ \ T_4 = 10 xx 3^3 = 270\ \ text{(True)}`

`text(In C,)\ \ T_6 = 10 xx 3^5 = 2430\ \ text{(NOT true)}`

`text(In D,)\ \ T_8 = 10 xx 3^7 = 21\ 870\ \ text{(True)}`

`text(In E,)\ \ T_10 = 10 xx 3^9 = 196\ 830\ \ text{(True)}`

`=> C`

PATTERNS, FUR1 2011 VCAA 2 MC

The first three terms of an arithmetic sequence are –3, –7, –11 . . .

An expression for the `n`th term of this sequence, `t_n`, is

A. `t_n = 1 - 4n`

B. `t_n = 1 - 8n`

C. `t_n = -3 - 4n`

D. `t_n = -3 + 4n`

E. `t_n = -7 + 4n`

Show Answers Only

`A`

Show Worked Solution

`text(Sequence is –3, –7, –11, …)`

`text(AP where)\ \ \ a`	`= –3, and`
`d`	`= –7 – (–3) = –4`

`t_n`	` = a + (n – 1) d`
	` = –3 + (n – 1) (–4)`
	` = –3 – 4n + 4`
	` = 1 – 4n`

`=> A`

PATTERNS, FUR1 2014 VCAA 8 MC

The first term of a geometric sequence is `a`, where `a < 0`.
The common ratio of this sequence, `r`, is such that `r < –1`.
Which one of the following graphs best shows the first 10 terms of this sequence?

Show Answers Only

`B`

Show Worked Solution

`text(By elimination)`

♦ Mean mark 40%.

`a < 0. text(Cannot be C.)`

`r < -1. text(Successive terms change sign and)`

`text(increase exponentially.)`

`text(Cannot be A, D, or E.)`

`=> B`

PATTERNS, FUR1 2014 VCAA 5 MC

Mary plans to read a book in seven days.
Each day, Mary plans to read 15 pages more than she read on the previous day.
The book contains 1155 pages.
The number of pages that Mary will need to read on the first day, if she is to finish reading the book in seven days, is

A. `112`

B. `120`

C. `150`

D. `165`

E. `180`

Show Answers Only

`B`

Show Worked Solution

`text(Sequence is)\ \ a, a+15, a + 2 xx 15, …`

`=>\ text(AP where)\ \ \d=15`

`text(Find)\ \ a\ \ text(when)\ S_7 = 1155`

`S_n`	`=n/2[2a + (n-1)d]`
`1155`	`=7/2[2a + (7-1)15]`
`1155`	`=7/2[2a + 90]`
	`=7a + 315`
`7a`	`=840`
`a`	`=120`

`=> B`

PATTERNS, FUR1 2014 VCAA 3 MC

A city has a population of 100 000 people in 2014.
Each year, the population of the city is expected to increase by 4%.
In 2018, the population is expected to be closest to

A. `108\ 000`

B. `112\ 000`

C. `115\ 000`

D. `117\ 000`

E. `122\ 000`

Show Answers Only

`D`

Show Worked Solution

`text(Population is a geometric sequence with)`

`a = 100\ 000,\ \ \ \ r = 1.04`

`P_1 (2014)`	`= a = text(100 000)`
`P_2 (2015)`	`= ar^1 = text(100 000) (1.04) = text(104 000)`
`vdots`
`P_5 (2018)`	`= ar^4 = text(100 000) (1.04)^4 = text(116 985.8…)`

`∴\ text(In 2018, the population will be closest)`

`text(to)\ 117\ 000.`

`=> D`

PATTERNS, FUR1 2014 VCAA 1-2 MC

Paul went running every morning from Monday to Sunday for one week.
On Monday, Paul ran 1.0 km.
On Tuesday, Paul ran 1.5 km.
On Wednesday, Paul ran 2.0 km.
The number of kilometres that Paul ran each day continued to increase according to this pattern.

Part 1

The number of kilometres that Paul ran on Thursday is

A. `2.5`

B. `3.0`

C. `3.5`

D. `4.0`

E. `5.0`

Part 2

The total number of kilometres that Paul ran during the week is given by

A. the seventh term of an arithmetic sequence with `a = 1` and `d = 0.5`

B. the seventh term of a geometric sequence with `a = 1` and `r = 0.5`

C. the sum of seven terms of an arithmetic sequence with `a = 1` and `d = 0.5`

D. the sum of seven terms of a geometric sequence with `a = 1` and `r = 0.5`

E. the sum of seven terms of a Fibonacci-related sequence with `t_1 = 1` and `t_2 = 1.5`

Show Answers Only

`text(Part 1:)\ A`

`text(Part 2:)\ C`

Show Worked Solution

`text(Part 1)`

`text(Sequence is 1.0, 1.5, 2.0, …)`

`a = 1.0\ \ \ \ d = 1.5-1.0 = 0.5`

`T_4 = text(Thursday’s distance)`

`T_4`	`= a + (n-1) d`
	`=1.0 + (4-1) 0.5`
	`=2.5`

`=> A`

`text(Part 2)`

`text (Series is an arithmetic series with)\ a = 1.0 and d = 0.5`

`:.\ text(Total kms run in week is the sum of the first)`

`text(seven terms of the arithmetic series.)`

`=> C`

PATTERNS, FUR1 2011 VCAA 1 MC

Stefan swam laps of his pool each day last week.

The number of laps he swam each day followed a geometric sequence.

He swam 1 lap on Monday, 2 laps on Tuesday and 4 laps on Wednesday.

The number of laps that he swam on Thursday was

A. `5`

B. `6`

C. `8`

D. `12`

E. `16`

Show Answers Only

`C`

Show Worked Solution

`text(Sequence is 1, 2, 4, …`

`text(GP where)\ \ \ a`	`= 1, and`
`r`	`= t_2 / t_1 = 2`

`T_n = ar^(n-1)`

`:. T_4\ text{(Thurs)}`	` = 1 xx 2^3`
	` = 8`

`=> C`

PATTERNS, FUR1 2010 VCAA 9 MC

Before he began training, Jethro’s longest jump was 5.80 metres.
After the first month of training, his longest jump had increased by 0.32 metres.
After the second month of training, his longest jump had increased by another 0.24 metres.
After the third month of training, his longest jump had increased by another 0.18 metres.

If this pattern of improvement continues, Jethro’s longest jump, correct to two decimal places, will be closest to

A. `6.54\ text(metres.)`

B. `6.68\ text(metres.)`

C. `7.00\ text(metres.)`

D. `7.08\ text(metres.)`

E. `7.25\ text(metres.)`

Show Answers Only

`D`

Show Worked Solution

`text(Sequence of “increases” is)`

♦ Mean mark 38%.

`0.32, 0.24, 0.18,\ …`

`text(GP where)\ \ \ a`	`= 0.32, and`
`r`	`=t_(2)/t_(1)=0.24/0.32=0.75`

`text(S)text(ince)\ \ \|\ r\ |< 1,`

`S_oo`	`=a/(1 – r)`
	`=0.32/(1 – 0.75)`
	`=1.28`

`:.\ text(Jethro’s longest jump will be 5.8 + 1.28 = 7.08 m.)`

`=> D`

PATTERNS, FUR1 2010 VCAA 8 MC

The `n`th term in a geometric sequence is `t_n`.

The common ratio is greater than one.

A graph that could be used to display the terms of this sequence is

Show Answers Only

`A`

Show Worked Solution

`text(GP where)\ \ r>1`

♦♦♦ Mean mark 16%!

`text(General term is)\ \ \ t_n=ar^(n – 1)`

`text(Consider)\ \a>0,`

`text(the graph of)\ \t_n\ \text(will be positive and)`

`text(increase exponentially, but no graph shows this.)`

`text(Consider)\ \a<0,`

`text(the graph will be negative and increase)`

`text(exponentially in a negative direction.)`

`=> A`

PATTERNS, FUR1 2010 VCAA 5 MC

A team of swimmers was training.
Claire was the first swimmer for the team and she swam 100 metres.
Every other swimmer in the team swam 50 metres further than the previous swimmer.
Jane was the last swimmer for the team and she swam 800 metres.

The total number of swimmers in this team was

A. `9`

B. `13`

C. `14`

D. `15`

E. `18`

Show Answers Only

`D`

Show Worked Solution

`text(Sequence is 100, 150, 200, … ,800)`

`text(AP where)\ \ \ a`	`= 100`
`d`	`=50`

`T_n`	`=a + (n – 1)d`
`800`	`=100 + (n – 1)50`
`700`	`=50n – 50`
`50n`	`=750`
`n`	`=15`

`=> D`

PATTERNS, FUR1 2012 VCAA 8 MC

The graph above shows consecutive terms of a sequence.

The sequence could be

A. geometric with common ratio `r`, where `r< 0`

B. geometric with common ratio `r`, where `0 < r < 1`

C. geometric with common ratio `r`, where `r > 1`

D. arithmetic with common difference `d`, where `d< 0`

E. arithmetic with common difference `d`, where `d> 0`

Show Answers Only

`B`

Show Worked Solution

`text (As)\ \ n\ \ text(increases,) \ \ t_n→0`

♦ Mean mark 50%.

`:.\ text (Sequence is a geometric with a limiting sum)`

`text(i.e.)\ \ |\ r\ | < 1`

`text (Only one choice satisfies these conditions.)`

`rArr B`

PATTERNS, FUR1 2012 VCAA 7 MC

A dragster is travelling at a speed of 100 km/h.

It increases its speed by

50 km/h in the 1st second
30 km/h in the 2nd second
18 km/h in the 3rd second

and so on in this pattern.

Correct to the nearest whole number, the greatest speed, in km/h, that the dragster will reach is

A. `125`

B. `200`

C. `220`

D. `225`

E. `250`

Show Answers Only

`D`

Show Worked Solution

`text (Sequence of speed increases is)`

`text (50, 30, 18, …)`

`text (GP where)\ \ \ a`	`= 50, and`
`r`	`= t_2/t_1 = 30/50 = 0.6`

`text (S) text (ince)\ \ |\ r\ | < |,`

`S_oo`	`= a / (1-r)`
	`= 50/ (1 – 0.6)`
	`= 125`

`:.\ text (Max speed is 100 + 125 = 225 km/h)`

`rArr D`

PATTERNS, FUR1 2012 VCAA 6 MC

The second and third terms of a geometric sequence are 100 and 160 respectively.

The sum of the first ten terms of this sequence is closest to

A. `4300`

B. `6870`

C. `11\ 000`

D. `11\ 290`

E. `11\ 350`

Show Answers Only

`E`

Show Worked Solution

`text (GP where)\ \ \ T_2 = 100, and T_3 = 160`

`:. r= T_3/T_2 = 160/100 = 1.6`

`T_2`	`= ar`
`100`	`= a xx 1.6`
`:. a`	`= 62.5`

`text (Find)\ \ S_10`

`S_n`	`= (a (r^n – 1))/ (r-1)`
`S_10`	`= (62.5 (1.6^10 – 1))/(1.6 – 1)`
	`=11\ 349.07…`

`rArr E`

PATTERNS, FUR1 2012 VCAA 5 MC

On the first day of a fundraising program, three boys had their heads shaved.

On the second day, each of those three boys shaved the heads of three other boys.

On the third day, each of the boys who was shaved on the second day shaved the heads of three other boys.

The head-shaving continued in this pattern for seven days.

The total number of boys who had their heads shaved in this fundraising activity was

A. `2187`

B. `2188`

C. `3279`

D. `6558`

E. `6561`

Show Answers Only

`C`

Show Worked Solution

`text (Sequence is 3, 9, 27, …)`

♦ Mean mark 48%.

`text (GP where)\ \ \ a`	`= 3, and`
`r`	`= t_2/t_1 = 9/3 = 3`

`S_7`	`=\ text (Total boys with head shaved)`
`S_n`	`= (a (r^n – 1))/(r-1)`
`:. S_7`	`= (3 (3^7 – 1))/(3-1)`
	`= 3279`

`rArr C`

PATTERNS, FUR1 2012 VCAA 3-4 MC

Use the following information to answer Parts 1 and 2.

As part of a savings plan, Stacey saved $500 the first month and successively increased the amount that she saved each month by $50. That is, in the second month she saved $550, in the third month she saved $600, and so on.

Part 1

The amount Stacey will save in the 20th month is

A. `$1450`

B. `$1500`

C. `$1650`

D. `$1950`

E. `$3050`

Part 2

The total amount Stacey will save in four years is

A. `$13\ 400`

B. `$37\ 200`

C. `$58\ 800`

D. `$80\ 400`

E. `$81\ 600`

Show Answers Only

`text (Part 1:)\ A`

`text (Part 2:)\ D`

Show Worked Solution

`text (Part 1)`

`text (Sequence is 500, 550, 600,…)`

`text (AP where)\ \ a`	`= 500, and`
`d`	`= text (550 – 500 = 50)`

`T_n`	`= a + (n – 1) d`
`T_20`	`= 500 + (20 – 1)50`
	`= 1450`

`rArr A`

`text (Part 2)`

`n`	`= 4 xx 12 = 48`
`S_n`	`= n/2 [2a + (n-1)d]`
`S_4`	`= 48/2 [2 xx 500 + (48-1)50]`
	`= 24 [1000 + 2350]`
	`= 80\ 400`

`rArr D`

PATTERNS, FUR1 2010 VCAA 4 MC

The first four terms of a geometric sequence are

`4, – 8, 16, – 32`

The sum of the first ten terms of this sequence is

A. `–2048`

B. `–1364`

C. `684`

D. `1367`

E. `4096`

Show Answers Only

`B`

Show Worked Solution

`4, –8, 16, –32)`

♦ Mean mark 41%.

`text(GP where)\ \ \ a`	`= 4`
`r`	`=t_(2)/t_(1)= (–8)/4=–2`

`S_n`	`=(a(r^n – 1))/(r – 1)`
`:.S_10`	`=[4[(–2)^10 – 1]]/(–2 – 1)`
	`= –1364`

`=> B`

PATTERNS, FUR1 2010 VCAA 3 MC

The prizes in a lottery form the terms of a geometric sequence with a common ratio of 0.95.
If the first prize is $20 000, the value of the eighth prize will be closest to

A. `$7000`

B. `$8000`

C. `$12\ 000`

D. `$13\ 000`

E. `$14\ 000`

Show Answers Only

`E`

Show Worked Solution

`text(GP has)\ \ r = 0.95`

`T_1`	`=a=20000`
`T_2`	`=ar=20000(0.95)=19\ 000`
`T_3`	`=ar^2=20000(0.95)^2=18\ 050`
`vdots`
`T_8`	`=ar^7=20000(0.95)^7=13\ 966.74`

`=> E`

PATTERNS, FUR1 2010 VCAA 2 MC

The first three terms of a geometric sequence are

`0.125, 0.25, 0.5`

The fourth term in this sequence would be

A. `0.625`

B. `0.75`

C. `0.875`

D. `1`

E. `1.25`

Show Answers Only

`D`

Show Worked Solution

`text(GP sequence is 0.125, 0.25, 0.5)`

`a`	`=0.125`
`r`	`=t_(2)/t_(1)=0.25/0.125=2`
`T_4`	`=ar^3`
	`=0.125 xx 2^3`
	`=1`

`=> D`

PATTERNS, FUR1 2010 VCAA 1 MC

The sequence

`3, 6, 9, 12\ . . .`

could be

A. Fibonacci.

B. arithmetic.

C. geometric.

D. alternating.

E. decreasing.

Show Answers Only

`B`

Show Worked Solution

`text(Sequence is 3, 6, 9, 12 . . .)`

`text(Common difference)\ =3`

`:.\ text(Arithmetic)`

`=> B`

PATTERNS, FUR1 2013 VCAA 6 MC

There are 3000 tickets available for a concert.

On the first day of ticket sales, 200 tickets are sold.

On the second day, 250 tickets are sold.

On the third day, 300 tickets are sold.

This pattern of ticket sales continues until all 3000 tickets are sold.

How many days does it take for all of the tickets to be sold?

A. `5`

B. `6`

C. `8`

D. `34`

D. `57`

Show Answers Only

`C`

Show Worked Solution

`200+250+300+…`

`text(AP where)\ \ \ a`	`=200`
`d`	`=250-200=50`

`text(Find)\ n,\ text(when)\ S_n=3000`

`S_n`	`=n/2[2a+(n-1)d]`
`3000`	`=n/2[2×200+(n-1)50]`
	`=n/2(400+50n-50)`
	`=n/2(350+50n)`
`3000`	`=175n+25n^2`

`n^2+7n-120`	`=0`
`(n-8)(n+15)`	`=0`

`:. n=8,\ \ \ (n>0)`

`=> C`

PATTERNS, FUR1 2013 VCAA 4 MC

The vertical distance, in m, that a hot air balloon rises in each successive minute of its flight is given by the geometric sequence

`64.0,\ \ 60.8,\ \ 57.76\ …`

The total vertical distance, in m, that the balloon rises in the first 10 minutes of its flight is closest to

A. `38`

B. `40`

C. `473`

D. `514`

E. `1280`

Show Answers Only

`D`

Show Worked Solution

`text(Series 64.0, 60.8, 57.76 …)`

`text(GP where)\ \ a`	`=64.0`
`r`	`=60.8/64.0=0.95`

`S_n`	`=(a(1-r^n))/(1−r)`
`S_10`	`=(64 (1−0.95^10))/(1−0.95)`
	`=513.616…`

`=> D`

PATTERNS, FUR1 2013 VCAA 3 MC

The first time a student played an online game, he played for 18 minutes.
Each time he played the game after that, he played for 12 minutes longer than the previous time.
After completing his 15th game, the total time he had spent playing these 15 games was

A. `186` minutes

B. `691` minutes

C. `1206` minutes

D. `1395` minutes

E. `1530` minutes

Show Answers Only

`E`

Show Worked Solution

`text(Series)\quad18, 18+12, 18+(2×12), …`

`text(AP with)\quad a=18 and d=12`

`S_n`	`=n/2[2a+(n-1)d]`
`S_15`	`=15/2[2(18)+(15-1)12]`
	`=15/2(36+168)`
	`=15/2(204)`
	`=1530`

`=> E`

PATTERNS, FUR1 2013 VCAA 2 MC

The graph above shows the first six terms of a sequence.

This sequence could be

A. an arithmetic sequence that sums to one.

B. an arithmetic sequence with a common difference of one.

C. a Fibonacci-related sequence whose first term is one.

D. a geometric sequence with an infinite sum of one.

E. a geometric sequence with a common ratio of one.

Show Answers Only

`E`

Show Worked Solution

`text(Series is 1, 1, 1, …)`

♦ Mean mark 41%.

`text(The only possibility within the choices)`

`text(is a geometric sequence where)\ \ r=1.`

`=> E`

PATTERNS, FUR1 2013 VCAA 1 MC

The first three terms of an arithmetic sequence are 3, 5 and 7.

The ninth term of this sequence is

A. `9`

B. `17`

C. `19`

D. `21`

E. `768`

Show Answers Only

`C`

Show Worked Solution

`text(AP is 3, 5, 7 …)`

`a=3,\ \ \ d=5-3=2`

`T_n`	`=a+(n-1)d`
`T_9`	`=3+(9-1)2`
	`=19`

`=> C`

`S_∞`	`= a/(1 – r)`
	`= 16/(1 – 0.75)`
	`= 64`

`t_7`	`= ar^6`
	`= 2 xx 3^6`
	`= 1458`

`S_n`	`= (a(r^n – 1))/(r – 1)`
`:. S_5`	`= (64(1.25^5 – 1))/(0.25)`
	`= 525.25`

`t_n`	`=ar^(n-1)`
`t_5`	`= 5 xx 2^4`
	`= 80`

`r`	`=x/6=54/x`
`x^2`	`=54 xx 6`
	`=324`
`:.x`	`=18`

`:.\ text(Max. height)`	`= 80 + 64`
	`= 144`

`S_n`	`=(a(1-r^n))/(1-r)`
`:.S_14`	`=(15(1-0.9^14))/(1-0.9)`
	`=115.68…`

`S_n`	`=n/2(a+l)`
`:.S_20`	`=20/2 (20+1)`
	`=210`

`T_4`	`=a+3d`
	`=15+3 xx 7`
	`=36`