Consider an arithmetic sequence 2,6,10,1iwi 0,tz dx; kv0stm274, $\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 sequence has j1-/oa5rl4krnzanf f pd (e)84yqx a.$u_{1}=40, u_{2}=32, u_{3}=24$ .
1. Find the common difference, d .   
2. Find $u_{8}$ .   
3. Find $S_{8}$ .   

Only one of the following four sequences is arithmetic and only t tve9sr(l(u40( r pivone of them is geometi l ttrv4(us0v(er9p( ric.

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

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 four sequences is a: qgxy7tt/p/7r+ cgpgflvi 20t k4 z0urithmetic and only one of them is geop/ktqz 2v fg 07yp4: +0ltictgr/ux7gmetric.

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

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 into a savi br1y o/bic-.x9 sd qlp190dizngs account that pays an annual interest rate of bypb1q9 ox9s z1-c /diidlr0 .5.5 % , compounded annually.
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 geometric s*5pxm .r2r4 e l0sma1lrh hnk(equence 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 given by q 4lxi 7r,ea74 x*ceqp 6z ha/z:e*gpf3,5,7, $\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,sc f+/7bsudg q8sa;f: 5vo:oj5760,-4608, $\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 ground n times. The heights of the bou ttow19 ttsg8*nces, $h_{1}, h_{2}, h_{3}, \ldots, h_{n}$ , 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 difft lda(6j-yur; erent sizes. The height of the ; y-rlu6j(dat largest 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 3 pt;tkgkhn+n wi6b;; $ 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 sequenceqjxy2k 91cdwb /s)iw. g+r; xu of figures.


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 sequence, -iluggue,*u65fm 6aq,q u7lk $u_{5}=24$, $u_{13}=80$ .
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 geometric seqk w 8vzww/.ns(,;kcsuuence 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 sequencd+gt6d 7kp;y ve, $u_{4}=12$, $u_{11}=-9$ .
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 t;m*ffzsz :wm lr.pace 2 p:)w/h term 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 sum of the 2 nd 6r7 dl6pp.9m fx rgcvu am2pkd;l-. /vand 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 45000 k v +fun.pia;2loalas left in the wild in 2019 . A year before, in 2018 , the population of koalas was estimated as 50000 . Assuming the population of koalas continues to decrease by the same lvp 2auf. ni;+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 London to Edinburgh an ffnuo 1+oe9pc,hl4)l7wd du(d the distance between each la,(woen 7 dfo4c+ )fpllud1 9uhndmark 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 sequence is 24 and the common differ*hz1p ja6ay* ulvyj 89ws6td*ence is 16 . ul9*w jyp d8j*a6*1z svayth 6
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 ah;pp zd89+m3gry lgk/ bank loan to purchase a new Tesla electric car. Fiona takes out a loan of l9yk;8/zg3+mhdrp g p P dollars with a bank that offers a nominal annual interest rate of 2.6 % , compounded monthly.
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 mw,4zfm 99 fsmnominal annual interest rate of 5 % , compounded monthly. The amount of money in Gary's account at the end of each year follows a geometric sequenf 9zs,w9mm 4fmce 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 a6z :h(.fmq8a 5xp ewt m8xma8woja n:.nd the ninth 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 sequence ara goet rc :il4(ra6c43;7uof de $ 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 anpf)igjj96 z7s(h q9c geometric sequence are $u_{1}=0.4, u_{2}=0.6, u_{3}=0.9$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 sequenc8ol q7/h if*ave, $u_{2}=6, u_{5}=20.25$ .
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 whole num8btr4mn5 *ycvsu-:w5zwhk) fber.
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) ocy/k*tfb*9 ; ojxf)5h oar/o pver the last six months. He decided to deposit his savings in a bank which offers a nominal annuoo*tfx p* 5hcyf)ak9b ;r j/o/al 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 a bank account. Thec *;cwcsr8 f3j bank pays a nominal annual interest rate of 4 % , compoundedc* rjscw8;cf 3 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 correct to two decjtjj5 ;ad uf50imal 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 74000 aa91gd);vgpxAustralian dollars (AUDp)ag1 x9;vad g).
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 forzz32e5ti lpx my( 46tk(nphr0 $ 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 , Grace invests P6jfnlonb*o0e 4xils:n 1yv75 dollars in an account that pays a nominal annual interest rate of 6 % , compounded quarterly. The amount of money in Grace's accoun6 0jn5n* :vls4nx7ybf e 1oiolt at the end of each year follows a geometric 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 ir1 ty2apw(90v35eiy0b n pgvlunch break. The height of the first swing wia(twb0vyypni5 92p g 13er v0as 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 first mp1pctq13vvw 5io y;euy,p4z 0 inute, 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 am. Will Sarah arrive to school on timvw ip 1zp ;5,3etuoyy4qvc10pe? Justify your answer.

Jack rides his bike to wgw5 ugc me:t.ji :.sk4ork each morning. During the first minute, he travels 160 metres. In5:kg.wmisju. etg4 :c each subsequent 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 arithmetijwj 5o7tr+e z,tz(0* sew7wvq c sequence is equal to 7 and the sum of the first 8 terms is 20 . Fi j7w*vtj zs75o reqtww,0+z (end the common difference and the first term.   

  The first term and the common ratio of a geometric wj; i5ogcb )v1os3bgl *juc g8i mb8f6j9o8;jseries are denoted, respectivelyj8og;o8 cij f 8sgjlu1b9bvo bwi3g6jm ;c*) 5, by $u_{1}$ 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 sequence is equal to j)q a5e*0:bjsomeb b 21 and the sum of the first 16 terms is 52 . Find the common difference and the first term. 0j 5abeoq2 b*b )j:mse$u_1$ =   

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

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

  The 1st, 5 th and 13 th terms :qg v7kn9nro(uk;e 0aof an arithmetic sequence, with common difference 7;kre n9(v an q0:uokgd, $d \neq 0$ , 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 willgusa ex;3w/n d1 9m:sw decrease by 5 % each year unle wuxgsm/:;end 39sa1w ss some new trees are planted. At the end of each year, 600 new trees are planted 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 k-4,4, k+2 , whek -w)snek2-at6u w h7wre hk) ek aw2wwtn 6u-s-7$k \in \mathbb{Z}$ .
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 saving 30000 dollars to put towards a jx ,( i 8g9x d:zbrtgj(upy09shouse deposit. They each have 16000 dollars to i(pzbugxrx(j:y 9 , gi ds9tj80nvest.
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 sng1ybrc8+3w q f 64ocj-jx+k pay for his college tuition. f+cjcy g6qbj ro-81xk 3 4+nswHe 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, in order, are .ls ,io a;5lp*2qhbd x8 tm+ cie0rxw4 $2 \ln x, q \ln x, \ln \sqrt{x}$ 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 squar jzokk+, te0 ut a;d;236rkktie are 8 $\mathrm{~cm}$ 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 infinite geo7pwrhsnfbzf61xog. j1.um 2 )metric 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 integers, between 20 and 300 , w*hlmbhp 4fp+dnl :8)wpsa/,/g mherr*- e 7 vmhich 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 geometric s r1cx9r dno q,q etf++tvpe.)nc )g4w*equence 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 infia em,))j otq/d8) haxgnite geometric 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;e2 w)4jobjt *z slb ldn5lm;+ premium electric car, at an annual interest rate of 5.49 % . The interest im+e; bln j)4to;5* zb2ljsdwl s calculated at the end of each year 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 properties of he;s i7q45v.6 wkoybq8jg q l*saxagonal 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 polynomial equati d;bpr mx)-vyg*82 vzaon $x^{3}+b x^{2}+c x+d=0$ 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 .