Consider an arithmetic se,x p 4d6*e kvc)cs9:7lvxshm jquence 2,6,10,14, $\ldots $
1. Find the common difference, d .   
2. Find the 10th term in the sequence.   
3. Find the sum of the first 10 terms in the sequence.   

  An arithmetic sequenc)eof.x hs3,f-imz-j ye has $u_{1}=40, u_{2}=32, u_{3}=24$ .
1. Find the common difference, d .   
2. Find $u_{8}$ .   
3. Find $S_{8}$ .   

Only one of the follow:g;v (u1 enwjlwo rv68ing four sequences is arithmetic and only one of them is geom(6 rv w1;jwnvgo8:eu letric.

$\begin{aligned}
a_{n} & =1,5,10,15, \ldots & c_{n} & =1.5,3,4.5,6, \ldots \\
b_{n} & =\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots & d_{n} & =2,1, \frac{1}{2}, \frac{1}{4}, \ldots
\end{aligned}$

1. State which sequence is arithmetic and find the common difference of the sequence.
2. State which sequence is geometric and find the common ratio of the sequence.
3. For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction.

Only one of the following forxvcbp5 )- 5rsur sequences is arithmetic and only one of them is geocvrsr 5b5)x-pmetric.

$\begin{aligned}
a_{n} & =\frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots & c_{n} & =3,1, \frac{1}{3}, \frac{1}{9}, \ldots \\
b_{n} & =2.5,5,7.5,10, \ldots & d_{n} & =1,3,6,10, \ldots
\end{aligned}$

1. State which sequence is arithmetic and find the common difference of the sequence.
2. State which sequence is geometric and find the common ratio of the sequence.
3. For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction.

Jeremy invests 8000 dollars in,3 scvn,yf4x w:ybps9to a savings account that pays an annual interest rate of 5.5 % , compounded annually. 39v, y,x sb wsp:ync4f
1. Write down a formula which calculates that total value of the investment after n years.
2. Calculate the amount of money in the savings account after:
1. 1 year;
2. 3 years.
3. Jeremy wants to use the money to put down a $ 10000 deposit on an apartment. Determine if Jeremy will be able to do this within a 5 -year timeframe.

  Consider the infinite geomkw3ob.7s f/q,g b qsah:tc 4m1etric sequence 4480,-3360,2520,-1890, ...
1. Find the common ratio, r .   
2. Find the 20 th term.≈   
3. Find the exact sum of the infinite sequence.   

  $\text { The table shows the first four terms of three sequences: } u_{n}, v_{n} \text {, and } w_{n} \text {. }$



1. State which sequence is
$u_{n}$ is A $v_{n}$ is B $w_{n}$ is C

1. arithmetic; =  ----待选择----ABC 
2. geometric. =  ----待选择----ABC 
2. Find the sum of the first 50 terms of the arithmetic sequence.   
3 . Find the exact value of the 13 th term of the geometric sequence.   

  An arithmetic sequence is given4t-rwv x,5 fzq/k6pjo by 3,5,7, $\ldots $
1. Write down the value of the common difference, d .   
2. Find
1. $u_{10}$ ;   
2. $S_{10}$=   
3. Given that $u_{n}=253$ , find the value of n .   

  Consider the infinite geometric sequence 9000,-7200,5760,-4608,/rixy6 ;7 k3hs9kh4bey,mbfcj1c zt+ $\ldots $
1. Find the common ratio.   
2. Find the 25 th term.≈   
3. Find the exact sum of the infinite sequence.   

  A tennis ball bounces on the grd:u( 4pk3 mg)tm 6wgbug w7x.hound n times. The heights of the bounces, $h_{1}, h_{2}, h_{3}, \ldots, h_{n}$ , form a geometric sequence. The height that the ball bounces the first time, $h_{1}$ , is 80 cm, and the second time, $h_{2}$ , is $60 \mathrm{~cm} $.
1. Find the value of the common ratio for the sequence.   
2. Find the height that the ball bounces the tenth time, $h_{10}$ .≈    cm
3. Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to 2 decimal places.≈    cm

  The third term, $u_{3}$ , of an arithmetic sequence is 7 . The common difference of the sequence, d , is 3 .
1. Find $u_{1}$ , the first term of the sequence.   
2. Find $u_{60}$ , the 60 th term of sequence.

The first and fourth terms of this arithmetic sequence are the first two terms of a geometric sequence.   
3. Calculate the sixth term of the geometric sequence.≈   

  The fifth term, $ u_{5}$ , of a geometric sequence is 125 . The sixth term, $u_{6}$ , is 156.25 .
1. Find the common ratio of the sequence. =   
2. Find $u_{1}$ , the first term of the sequence. =   
3. Calculate the sum of the first 12 terms of the sequence.≈   

  The fourth term, $u_{4}$ , of a geometric sequence is 135 . The fifth term, $u_{5}$ , is 81 .
1. Find the common ratio of the sequence. =   
2. Find $u_{1}$ , the first term of the sequence. =   
3. Calculate the sum of the first 20 terms of the sequence.≈   

  The fifth term, $u_{5}$ , of an arithmetic sequence is 25 . The eleventh term, $u_{11}$ , of the same sequence is 49 .
1. Find d , the common difference of the sequence.   
2. Find $u_{1}$ , the first term of the sequence.   
3. Find $S_{100}$ , the sum of the first 100 terms of the sequence.   

  A 3D printer builds a set of 49 Eiffel Tower Replicas in differ gqn2s,kf.o2.ml g.xe ent sizes. The height of the largestk2l gq.2es ,gnxm.f. o tower in this set is $ 64 \mathrm{~cm}$ . The heights of successive smaller towers are 95 % of the preceding larger tower, as shown in the diagram below.



1. Find the height of the smallest tower in this set.≈    cm
2. Find the total height if all 49 towers were placed one on top of another.≈    cm

  Hannah buys a car for 4 or:uz-)d(l j lop6el $ 24900 . The value of the car depreciates by 16 % each year.
1. Find the value of the car after 10 years.

Patrick buys a car for 12000 dollars. The car depreciates by a fixed percentage each year, and after 6 years it is worth 6200 dollars . ≈   
2. Find the annual rate of depreciation of the car. ≈    %

Consider the following sequence of vz 4/-czhl3 nufigures.


Figure 1 contains 6 line segments.
1. Given that Figure n contains 101 line segments, show that n=20 .
2. Find the total number of line segments in the first 20 figures. __

  In an arithmetic sequeno o5x 6o)x0x8kqi*0f ni wogp1ce, $u_{5}=24$, $u_{13}=80$ .
1. Find the common difference.   
2. Find the first term.   
3. Find the sum of the first 20 terms in the sequence.   

  The first three terms of a geometrx zu qlm *;cwd5x (;.amyl35:lx 5fpqsic sequence are $u_{1}=32$, $u_{2}=-16$,$ u_{3}=8$ .
1. Find the value of the common ratio, r .   
2. Find $u_{6}$ . =   
3. Find $S_{\infty}$ . =   

  In an arithmetic sequence,ofisd*+ 1v:pohg 2q9 t $u_{4}=12$, $u_{11}=-9$ .
1. Find the common difference.   
2. Find the first term.   
3. Find the sum of the first 11 terms in the sequence.   

  In an arithmetic sequence, the sum of the 2 nd and 6 th terdgm 3v5ip/ wx1m is 32 .
Given that the sum of the first six terms is 120 , determine the first term and common difference of the sequence. $u_1$ =    d =   

  In an arithmetic sequence, the sumvwp) 3lq :im449y u.8wk.efoey g m;xqv0nf.j of the 2 nd and 6 th term is 32 .
Given that the sum of the first six terms is 120 , determine the first term and common difference of the sequence.An arithmetic sequence has first term 45 and common difference -1.5 .
1. Given that the k th term of the sequence is zero, find the value of k .

Let $S_{n}$ denote the sum of the first n terms of the sequence.   
2. Find the maximum value of $S_{n}$ .   

The Australian Koala Foundation estimates that there are about 45 y)d8xtcq g y /gs/z)4g iko+-:eh2xps000 koalas left in the wild in 2019 . A year before, in hox:-cgyk)qez/xp s s8id )/+t2gg y4 2018 , the population of koalas was estimated as 50000 . Assuming the population of koalas continues to decrease by the same percentage each year, find:
1. the exact population of koalas in 2022 ;
2. the number of years it will take for the koala population to reduce to half of its number in 2018 .

  Landmarks are placed along the road from Lod:/: vb gna 6kynt:sk5ndon to Edinburgh and the distance between each landmarvgnb:/d:ka kn:y st5 6k is 16.1 $\mathrm{~km}$ . The first landmark placed on the road is 124.7 $\mathrm{~km}$ from London, and the last landmark is near Edinburgh. The length of the road from London to Edinburgh is 667.1 $\mathrm{~km} $.
1. Find the distance between the fifth landmark and London.   
2. Determine how many landmarks there are along the road.   

The first term of an arithmetic sequeni7s k)+b9zdmv ce is 24 and the common difference is 16 . +vsdkmb97z)i
1. Find the value of the 62 nd term of the sequence.

The first term of a geometric sequence is 8 . The 4 th term of the geometric sequence is equal to the 13 th term of the arithmetic sequence given above. __
2. Write down an equation using this information. __
3. Calculate the common ratio of the geometric sequence. __

  On 1st of January 2021, Fiona decides to take out a bank loan to purchase aair g v7+a4tl )v;jx6c new Tesla electric car. Fiona takes out a loan of P dollars with a bank that offers a nominal annual interest rate of 2.6 % , compounded mx+jg64 7vi;a lr)t acvonthly.
The size of Fiona's loan at the end of each year follows a geometric sequence with common ratio, $\alpha$ .
1. Find the value of $\alpha$ , giving your answer to five significant figures.

The bank lets the size of Fiona's loan increase until it becomes triple the size of the original loan. Once this happens, the bank demands that Fiona pays the entire amount back to close the loan.≈   
2. Find the year during which Fiona will need to pay back the loan.   

  On Gary's 50 th birthday, he invests P dollars in an account that pays a nomi:/4nkil0;szo4bj v 7 qhmkm,vnal annual interest rate of 5 % , compounded monthly. The amount of money in Gary's account at the end of each year follows a geometmm0v7;4k/vil4sz qn,:hj kob ric sequence with common ratio, $\alpha$ .
1. Find the value of $\alpha$ , giving your answer to four significant figures.

Gary makes no further deposits or withdrawals from the account.≈   
2. Find the age Gary will be when the amount of money in his account will be double the amount he invested.   

  In an arithmetic sequence, the third term is 41 and the nint3qp/ p1nua pe (2bkri8vq.ma 9h term is 23 .
1. Find the common difference.   
2. Find the first term.   
3. Find the smallest value of n such that $S_{n}<0$ .   

  The first three terms of a geometric sequenc28i,mrbg/px h;smd-p e are $ u_{1}=0.8, u_{2}=2.4, u_{3}=7.2 $.
1. Find the value of the common ratio, r .   
2. Find the value of $S_{8}$ .   
3. Find the least value of n such that $S_{n}>35000$ .   

  The first three terms of a geometric sequence are:a7x-ywa msh-nov67t $u_{1}=0.4, u_{2}=0.6, u_{3}=0.9$a .
1. Find the value of the common ratio, r .≈    !num!2%
2. Find the sum of the first ten terms in the sequence.   
3. Find the greatest value of n such that $S_{n}<650$ .   

  In a geometric sequenxm78fy3 w wrd6ce, $u_{2}=6, u_{5}=20.25$ .
1. Find the common ratio, r .   
2. Find $u_{1}$ .   
3. Find the greatest value of n such that $u_{n}<200$ .   

  In this question give all answers correct to the nearest whs,,k2pxs9 ivn ole number.
A population of goats on an island starts at 232 . The population is expected to increase by 15 \% each year.
1. Find the expected population size after:
1. 10 years;≈   
2. 20 years.≈   
2. Find the number of years it will take for the population to reach 15000 .   

Maria invests $ 25000 into a savings account that pays a nominal annual interest rate of 4.25 % , compounded monthly.
1. Calculate the amount of money in the savings account after 3 years.
2. Calculate the number of years it takes for the account to reach 40000 dollars.

  Greg has saved 2000 British pounds (GBP) over the last si,oae2sh)zs0f;b* hl j x months. He decided to deposit his savings in a bank wso,a *2hf0j;b )ehszlhich offers a nominal annual interest rate of 8 % , compounded monthly, for two years.
1. Calculate the total amount of interest Greg would earn over the two years. Give your answer correct to two decimal places.

Greg would earn the same amount of interest, compounded semi-annually, for two years if he deposits his savings in a second bank.   
2. Calculate the nominal annual interest rate the second bank offers.≈   

Emily deposits 2000 Australian dollars (AUD) into n ,i5by ci.p6ma bank account. The bank pays a nominal annual interest rate of 4 % , comp nmic 65.iyb,pounded monthly.
1. Find the amount of money that Emily will have in her bank account after 5 years. Give your answer correct to two decimal places.

Emily will withdraw the money back from her bank account when the amount reaches 3000 AUD.
2. Find the time, in months, until Emily withdraws the money from her bank account.

  In this question give all answers corroy: 5fo).edt, b mxz0qq2u2 liect to two decimal places.
Mia deposits 4000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 6 % , compounded semi-annually.
1. Find the amount of interest that Mia will earn over the next 2.5 years.

Ella also deposits AUD into a bank account. Her bank pays a nominal annual interest rate of 4 % , compounded monthly. In 2.5 years, the total amount in Ella's account will be 4000 AUD.≈   
2. Find the amount that Ella deposits in the bank account.≈   

  Julia wants to buy a house that requires a deposit of 7400ih,k4zoc fn*00 Australian dollars (AUD)kn* i4hc,zo0 f.
Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of 5.5 % , compounded monthly.
1. Find the amount of AUD Julia needs to invest to reach 74000 AUD after 8 years. Give your answer correct to the nearest dollar.

Julia's parents offer to add 5000 AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of 3.5 % , compounded quarterly. ≈   
2. Find the number of years it would take Julia to save the 74000 AUD if she accepts her parents money and follows their advice. Give your answer correct to the nearest year.   

  Ali bought a car for ; yif km9h .jx(-ltor 4x*n2bi$ 18000 . The value of the car depreciates by 10.5 % each year.
1. Find the value of the car at the end of the first year.≈   
2. Find the value of the car after 4 years.≈   
3. Calculate the number of years it will take for the car to be worth exactly half its original value. ≈   

  On 1st of January 2022 , Grad cl5hyy; nsez :7m34vce invests P dollars in an account that pays a nominal annual interest rate of 6 % , compounded quarterly. The amount of money in Grace's account at the end of each year follows a geometlys5z;47 vyd ehn:cm3ric sequence with common ratio, $\alpha$ .
1. Find the value of $\alpha$ , giving your answer to four significant figures.

Grace makes no further deposits or withdrawals from the account.≈   
2. Find the year in which the amount of money in Grace's account will become triple the amount she invested.    years

Let $u_{n}=5 n-1$ , for $n \in \mathbb{Z}^{+}$ .
1. 1. Using sigma notation, write down an expression for $u_{1}+u_{2}+u_{3}+\cdots+u_{10}$ .
2. Find the value of the sum from part (a) (i).

A geometric sequence is defined by $v_{n}=5 \times 2^{n-1}$ , for $n \in \mathbb{Z}^{+}$ .
2. Find the value of the sum of the geometric series $\sum_{k=1}^{6} v_{k}$ .

  Peter is playing on a swing during a school lunch break. The height of the ;3b)j3nx j fgi +mdxt/*f9wcmfirst /3+gj3fd jt9wimm n *fb)cxx ;swing was 2 $\mathrm{~m}$ and every subsequent swing was 84 % of the previous one. Peter's friend, Ronald, gives him a push whenever the height falls below $1 \mathrm{~m}$ .
1. Find the height of the third swing.≈   
2. Find the number of swings before Ronald gives Peter a push. n =   
3. Calculate the total height of swings if Peter is left to swing until coming to rest.   

Sarah walks to school each morning. During the f)x 12 n xnh xn:nd:,tpwofi90lirst minute, she travels 130 metres. In each subsequent minute, she travels 5 metres less than the distance she travelled during the previous minute. The distance from her home to school is 950 metres. Sarah leaves her house at 8: 00 am and must be at school by 8: 10 p n2x x1xhld0ti9nfnn :o),w:am. Will Sarah arrive to school on time? Justify your answer.

Jack rides his bike to work each morning. During the first minute, he travels 16 2jy ) vi+p,ng;+fnssl,ze *le0 metres. In each subsequejn * psl yzn+e;ise2lg f+,v,)nt minute, he travels 80 % of the distance travelled during the previous minute. The distance from his home to work is 750 metres. Jack leaves his house at 8:30 am and must be at work at 8:40 am. Will Jack arrive to work on time? Justify your answer.

  The third term of an arithmetic sequence is equal to 7 and the sum of th 4eyw9e(cp .khe first 8 terms is 20 . Find the common difference and thepyhe.9 kc(we 4 first term.   

  The first term and the common ratio of a geom9fq-di jxwhme1..rjkb;z5b 0etric series are denoted, respectively, bye1fq9rzb-hw0bmj ; . xk.di 5j $u_{1}$ and r , where $u_{1}$, $r \in \mathbb{Q}$ . Given that the fourth term is 64 and the sum to infinity is 625 , find the value of $u_{1}$ and the value of r .$u_{1}$ =    r =   

  The seventh term of an arithmetic sequensk( 8;44npjfumwl f -uce is equal to 1 and the sum of the first 16 terms is 52 . Find the common difference and the firsu4pklm8u f4(f;- swnjt term. $u_1$ =   

  $\text { The sum of an infinite geometric sequence is } 27 \text {. The second term of the sequence is } 6 \text {. Find the possible values of } r \text {. }$      

  $\text { The sum of the first three terms of a geometric sequence is } 92.5 \text {, and the sum of the infinite sequence is } 160 \text {. Find the common ratio. }$   

  The 1st, 5 th and 13 th terms of an ariti+ wu9al fky-;uz 4j2*ex2acf 3-hf rahmetic sequence, with common difference f e;uk2 ufhl-2acxfrz a+j3w49 -yia* d, $d \neq 0$ , are the first three terms of a geometric sequence, with common ratio r, $r \neq 1$ . Given that the 1 st term of both sequences is 12 , find the value of d and the value of r . r =    d =   

  $\text { The sum of the first three terms of a geometric sequence is } 81.3 \text {, and the sum of the infinite sequence is } 300 \text {. Find the common ratio. }$   

  It is known that the number of trees in a small forest will decrease by 5 %;5v yg.h (k6q u.ivzzf each year unless some new trees are planted. At the end of each year, 600 new trees are plan u6(fig5vh.y;z kzq .vted to the forest At the start of 2021 , there are 8200 trees in the forest.
1. Show that there will be roughly 9060 trees in the forest at the start of 2026 . ≈   
2. Find the approximate number of trees in the forest at the start of 2041 .≈   

  $\text { 1. The following diagram shows }[\mathrm{PQ}] \text {, with length } 4 \mathrm{~cm} \text {. The line is divided into an infinite number of line segments. The diagram shows the first four segments. }$




$\text { 1. The following diagram shows }[\mathrm{PQ}] \text {, with length } 4 \mathrm{~cm} \text {. The line is divided into an infinite number of line segments. The diagram shows the first four segments. }$The length of the line segments are m $\mathrm{~cm}$, $m^{2} \mathrm{~cm}, m^{3} \mathrm{~cm}, \ldots ,$ where $0\lt n \lt1$ .
Show that $m=\frac{4}{5}$ .   
2. The following diagram shows [RS], with length l $\mathrm{~cm}$ , where l$\gt $1 . Squares with side lengths $n \mathrm{~cm}, n^{2} \mathrm{~cm}, n^{3} \mathrm{~cm}, \ldots$ , where $0\lt n \lt1$ , are drawn along [RS]. This process is carried on indefinitely. The diagram shows the first four squares.


$\text { The total sum of the areas of all the squares is } \frac{25}{11} \text {. Find the value of } l \text {. }$   

The first three terms of an infinite geometric sequence are 0s l)v:b1g ltzg0a+gt k-4,4, k+2 , where sg g+ t0t) :bgv0za1ll$k \in \mathbb{Z}$ .
1. 1. Write down an expression for the common ratio, r .
2. Hence show that k satisfies the equation $k^{2}-2 k-24=0$ .
2. 1. Find the possible values for k .
2. Find the possible values for r .
3. The geometric sequence has an infinite sum.
1. Which value of r leads to this sum. Justify your answer.
2. Find the sum of the sequence.

  Let $f(x)=e^{3 \sin \left(\frac{\pi x}{4}\right)}$ , for x>0 .
The k th maximum point on the graph of f has x -coordinate $x_{k}$ , where k $\in \mathbb{Z}^{+}$ .
1. Given that $x_{k+1}=x_{k}+d$ , find d .   
2. Hence find the value of n such that $\sum_{k=1}^{n} x_{k}=992$ .   

Alex and Julie each have a goal of sa w*h9x:zwo9 i+s+obhzving 30000 dollars to put towards a house d:+z+ w*whibo9sx9zo heposit. They each have 16000 dollars to invest.
1. Alex chooses his local bank and invests his 16000 dollars in a savings account that offers an interest rate of 5 % per annum compounded annually.
1. Find the value of Alex's investment after 7 years, to the nearest hundred dollars.
2. Alex reaches his goal after n years, where n is an integer. Determine the value of n .
2. Julie chooses a different bank and invests her 16000 dollars in a savings account that offers an interest rate of r % per annum compounded monthly, where r is set to two decimal places.
Find the minimum value of r needed for Julie to reach her goal after 10 years.
3. Xavier also wants to reach a savings goal of 30000 dollars. He doesn't trust his local bank so he decides to put his money into a safety deposit box where it does not earn any interest. His system is to add more money into the safety deposit box each year. Each year he will add one third of the amount he added in the previous year.
1. Show that Xavier will never reach his goal if his initial deposit into the safety deposit box is 16000 dollars.
2. Find the amount Xavier needs to initially deposit in order to reach his goal after 7 years. Give your answer to the nearest dollar.

Grant wants to save 40000 dollars over 5 years to help his son pay for tx o5/bzk)7j0hjw jzwq*l6gv 2 kz,.ihis colleg z7 koq*/v26 wxk.iz,jt5) h0jgbwzlje tuition. He deposits 20000 dollars into a savings account that has an interest rate of 6 % per annum compounded monthly for 5 years.
1. Show that Grant will not be able to reach his target.
2. Find the minimum amount, to the nearest dollar, that Grant would need to deposit initially for him to reach his target.

Grant only has 20000 dollars to invest, so he asks his sister, Caroline, to help him accelerate the saving process. Caroline is happy to help and offers to contribute part of her income each year. Her annual income is 37500 dollars per year. She starts by contributing one fifth of her annual income, and then decreases her contributions by half each year until the target is reached. Caroline's contributions do not yield any interest.
3. Show that Grant and Caroline together can reach the target in 5 years.

Grant and Caroline agree that Caroline should stop contributing once she contributes enough to complement the deficit of Grant's investment.
4. Find the whole number of years after which Caroline will will stop contributing.

The first three terms of an infinite sequence+-k2sf;j/t0 ue/u nd c;oaaoc, in order, are $2 \ln x, q \ln x, \ln \sqrt{x}$ where x>0 .
First consider the case in which the series is geometric.
1. 1. Find the possible values of q .
2. Hence or otherwise, show that the series is convergent.
2. Given that q>0 and $ S_{\infty}=8 \ln 3$ , find the value of x .

Now suppose that the series is arithmetic.
3. 1. Show that $q=\frac{5}{4}$ .
2. Write down the common difference in the form $m \ln x$ , where $m \in \mathbb{Q}$ .
4. Given that the sum of the first n terms of the sequence is $\ln \sqrt{x^{5}}$ , find the value of n .

  The sides of a square are 8 x.f jsws/u)ln9:/ft+7wh qtb $\mathrm{~cm}$ long. A new square is formed by joining the midpoints of the adjacent sides and two of the resulting triangles are shaded as shown. This process is repeated 5 more times to form the right hand diagram below.




1. Find the total area of the shaded region in the right hand diagram above.≈   
2. Find the total area of the shaded region if the process is repeated indefinitely.≈   

The first two terms of an in lwj50)fp:fyucw 6s,zfinite geometric sequence, in order, are

$3 \log _{3} x, 2 \log _{3} x, \text { where } x>0 \text {. }$

1. Find the common ratio, r .
2. Show that the sum of the infinite sequence is $9 \log _{3} x$ .

The first three terms of an arithmetic sequence, in order, are

$\log _{3} x, \log _{3} \frac{x}{3}, \log _{3} \frac{x}{9}, \text { where } x>0$ .

3. Find the common difference d , giving your answer as an integer.

Let S_{6} be the sum of the first 6 terms of the arithmetic sequence. __
4. Show that $S_{6}=6 \log _{3} x-15$
5. Given that $S_{6}$ is equal to one third of the sum of the infinite geometric sequence, find x , giving your answer in the form $a^{p}$ where a, $p \in \mathbb{Z}$ .

Given a sequence of intege-azq;b3ft ya,f tn16brs, between 20 and 300 , which are divisible by 9 .
1. Find their sum.
2. Express this sum using sigma notation.

An arithmetic sequence has first term -500 and common difference of 8 . The sum of the first n terms of this sequence is negative.
3. Find the greatest value of n .

  The first three terms of a geomet ma4a* d(tbjw5 4j*g5 (oeiqkcric sequence are $\ln x^{9}, \ln x^{3}, \ln x$ , for x>0 .
1. Find the common ratio.   
2. Solve $\sum_{k=1}^{\infty} 3^{3-k} \ln x=27$ . $a^b$ a =    b =   

The first two terms of an infinite geo*u by,y 455e)pu-dig bcx:yme,py l0wmetric sequence are $u_{1}=20$ and $u_{2}=16 \sin ^{2} \theta$ , where $0<\theta<2 \pi$ , and $\theta \neq \pi$ .
1. 1. Find an expression for r in terms of $\theta$ .
2. Find the possible values of r .
2. Show that the sum of the infinite sequence is $\frac{100}{3+2 \cos 2 \theta}$ .
3. Find the values of $\theta$ which give the greatest value of the sum.

Bill takes out a bank loan of 100000 dollars to buy a premium electrd oiz3.x c7-fjau n:gzb)s/ y.ic car, at an annual interest rate of 5.49 % . The interest is calculated at the end of each yea.uonzsgizbc: -7yda/)x3 fj .r and added to the amount outstanding.
1. Find the amount of money Bill would owe the bank after 10 years. Give your answer to the nearest dollar.

To pay off the loan, Bill makes quarterly deposits of P dollars at the end of every quarter in a savings account, paying a nominal annual interest rate of 3.2 % . He makes his first deposit at the end of the first quarter after taking out the loan.
2. Show that the total value of Bill's savings after 10 years is $P\left[\frac{1.008^{40}-1}{1.008-1}\right]$ .
3. Given that Bill's aim is to own the electric car after 10 years, find the value for P to the nearest dollar.

Melinda visits a different bank and makes a single deposit of Q dollars, the annual interest rate being 3.5 % .
4. 1. Melinda wishes to withdraw 8000 dollars at the end of each year for a period of n years. Show that an expression for the minimum value of Q is

$\frac{8000}{1.035}+\frac{8000}{1.035^{2}}+\frac{8000}{1.035^{3}}+\cdots+\frac{8000}{1.035^{n}}$ .

2. Hence, or otherwise, find the minimum value of Q that would permit Melinda to withdraw annual amounts of 8000 dollars indefinitely. Give your answer to the nearest dollar.

This question asks you to investigate some prf0fv8q fd74p t1ir5ltoperties of hexagonal numbers.
Hexagonal numbers can be represented by dots as shown below where $h_{n}$ denotes the n th hexagonal number, $n \in \mathbb{N}$ .


Note that 6 points are required to create the regular hexagon $h_{2}$ with side of length 1 , while 15 points are required to create the next hexagon $h_{3}$ with side of length 2 , and so on.
1. Write down the value of $h_{5}$ .
2. By examining the pattern, show that $ h_{n+1}=h_{n}+4 n+1, n \in \mathbb{N} $.
3. By expressing $h_{n}$ as a series, show that $h_{n}=2 n^{2}-n, n \in \mathbb{N}$ .
4. Hence, determine whether 2016 is a hexagonal number.
5. Find the least hexagonal number which is greater than 80000 .
6. Consider the statement:
45 is the only hexagonal number which is divisible by 9 .
Show that this statement is false.

Matt claims that given $h_{1}=1 $ and $h_{n+1}=h_{n}+4 n+1, n \in \mathbb{N}$ , then

$h_{n}=2 n^{2}-n, \quad n \in \mathbb{N}$

7. Show, by mathematical induction, that Matt's claim is true for all $ n \in \mathbb{N}$ .

The cubic polynomialn+nse8 n. qp2 h/m*gxr equation $x^{3}+b x^{2}+c x+d=0$ has three roots $x_{1}$, $x_{2}$ and $x_{3}$ . By expanding the product $\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right) $, show that
1. 1. $ b=-\left(x_{1}+x_{2}+x_{3}\right)$ ;
2. $c=x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}$ ;
3. $d=-x_{1} x_{2} x_{3}$ .

It is given that b=-9 and c=45 for parts (b) and (c) below.
2. 1. In the case that the three roots $x_{1}$, $x_{2}$ and $x_{3}$ form an arithmetic sequence, show that one of the roots is 3 .
2. Hence determine the value of d .
3. In another case the three roots form a geometric sequence. Determine the value of d .