Only one of the following fove a2*j-jfx:eur sequences is arithmetic and only one of them is gee j*fj -:exv2aometric.


1.State which sequence is arithmetic and find the common difference of the sequence.
Hence C$_n$ is arithmetic and d=  .
2.State which sequence is geometric and find the common ratio of the sequence.
Hence d$_n$ is geometric and r=  .
3.For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction.
Using the nth term formula d$_n$= ----待选择----1/641/81/2 .

  Only one of the following four sequences is arithmetic and only one of-162xm+ j4up u3promptw tj-m them is geometric. tmrm--u6p4+mtxou jw 2p1jp3

1.State which sequence is arithmetic and find the common difference of the sequence.
Hence b$_n$ is arithmetic and d=  .
2.State which sequence is geometric and find the common ratio of the sequence.
Hence C$_n$ is geometric and r= ----待选择----1.531/3 .
3.For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction.

  An arithmetic sequence has ub rm)c,,b3 n7)sz 8 qmnxrnjb1$_1$=40,u$_2$=30,u$_3$=24

1.Find the common difference, d.
The common difference is d=-  .
2.Find u$_8$.
​Using the nth term formula u$_n$=-  .
3.Find S$_8$.
S$_8$=  .

  An arithmetic sequen7jd1aqng ao;9ce has u$_1$=12, u$_2$=21, $u_3$=30.
1.Find the common difference, d.
d=  .
2.Find $u_{10}$.
$u_{10}$=  .
3.Find $S_{10}$.
$S_{10}$=  .

  The 15th term of an arithmetic se *dfs w0t2w.enyh 0+p 7*sb.bip ephw9quence is 21 and the common difference is −4.

Find the first term of the sequence.
$u_1$=  .
Find the 29th term of the sequence.
$u_29$=-  .
Find the sum of the first 40 terms of the sequence.
$S_{40}$=-  .

  A geometric sequence has nwi1j1k 9as67yiiiu0 dvm 32n$u_1$=5, u$_2$=-1 and u$_3$=$\frac{1}{5}$.
Find the common ratio, r.
$r = \frac{-1}{x}$,那么$x$ =   .
Find the exact value of u$_7$.
$u_7 = \frac{1}{x}$,那么$x$ =   .
Find the exact value of $S_7$.
$S_7 = \frac{13021}{x}$,那么$x$ =   .

  A tennis ball bounces giyv1 twtoqs 64 ,7e3:76dqnfphe: gbon the ground n times. The heights of the bounces, ​$h_1,h_2,h_3,....,h_n$, 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 cm.

Find the value of the common ratio for the sequence.
r=  .
Find the height that the ball bounces the tenth time, $ℎ_10$​ .
$ℎ_10$​=  .
Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to 2 decimal places.
The number=  cm

  A geometric sequence has 20 terms, with ttcjgpp(zp j* ew9ysnejwo czx93yx+0,,;/ d-he first four terms given below.

418.5,279,186,124,…
​

1.Find r, the common ratio of the sequence. Give your answer as a fraction.
$r = \frac{2}{x}$ 那么$x$ =   
2.​Find $u_5$, the fifth term of the sequence. Give your answer as a fraction.
$u_5= \frac{x}{3}$ 那么$x$ =   
3.Find the smallest term in the sequence that is an integer.
$u_4$=  .
4.Find $S_10$ , the sum of the first 10 terms of the sequence. Give your answer correct to one decimal place.
$S_10$=  .

  Emily starts reading Leo Tolstoy's War and Peacx 6ubm:sol2bvomyhkjl5oq)9 * ) j8b8e on the 1st of February. The number ofmsj ul)xoybh9v8l5o*)628:bmb jq o k pages she reads each day increases by the same number on each successive day.

1.Calculate the number of pages Emily reads on the 14th of February.
$u_{14}$=  pages
2.Find the exact total number of pages Emily reads in the 28 days of February.
$S_{28}$=  pages

  The table shows the first four termg -i/xxjnm rxls4,)(es of three sequences: $u_n$ , $v_n$ , and $w_n$.

1.State which sequence is
1.1.arithmetic is  ----待选择----$u_n$$v_n$$w_n$ ;

1.2.geometric is  ----待选择----$u_n$$v_n$$w_n$ .

2.Find the sum of the first 50 terms of the arithmetic sequence.
$S_{50}$=  .
3.Find the exact value of the 13th term of the geometric sequence

$W_{13}$=  .

  The table below shows the first four terms of three sedwy6 evj66 1,7o-itl.g pt tphquences: $u_n , v_n , w_n$.


1.State which sequence is

1.1.arithmetic is  ----待选择----$u_n$$v_n$$w_n$ ;

1.2.geometric is  ----待选择----$u_n$$v_n$$w_n$ .

2.Find the exact value of the sum of the first 35 terms of the arithmetic sequence.
$S_{35}$=  .
3.Find the exact value of the 10th term of the geometric sequence.
$w_{10}$=  .

  The third term, $u_3$ , of an arithmetic sequence is 7. The common difference of
the sequence, d, is 3.
Find $u_1$ , the first term of the sequence.
$u_1$=  .
Find $u_{60}$ , the 60th term of sequence.
$u_{60}$=  .
The first and fourth terms of this arithmetic sequence are the first two terms
of a geometric sequence.

Calculate the sixth term of the geometric sequence.
$u_6$=  .

  The fifth term, $u_5$ , of a geometric sequence is 125. The sixth term, $u_6$ , is 156.25.

Find the common ratio of the sequence.
r=  .
Find $u_1$ , the first term of the sequence.
$u_1$=  .
Calculate the sum of the first 12 terms of the sequence.
$S_{12}$=  .

  The fourth term, $u_4$ , of a geometric sequence is 135. The sixth term, $u_5$ , is 81.

Find the common ratio of the sequence.
r=  .
Find $u_1$ , the first term of the sequence.
$u_1$=  .
Calculate the sum of the first 20 terms of the sequence.
$S_{20}$=  .

  The fifth term, $u_5$ , of a geometric sequence is 25. The eleventh term, $u_{11}$ , of the same sequence is 49.

Find d, the common difference of the sequence.
d=  .
Find $u_1$ , the first term of the sequence.
$u_1$=  .
Find $S_{100}$ ,the sum of the first 100 terms of the sequence.
$S_{100}$=  .

  The fifth term, $u_5$ , of a geometric sequence is 5. The eleventh term, $u_8$ , of the same sequence is 14.

Find d, the common difference of the sequence.
d=  .
Find $u_1$ , the first term of the sequence.
$u_1$=-  .
Find $S_{100}$ ,the sum of the first 100 terms of the sequence.
$S_{100}$=  .

  On the first day of September, 2019, Gloria planted 5 floo 1;2ot19grfct3gcsa a5c7o , usd h:rwers in her garden. The number of flowers, which she plants at every day of the month, forms an arithmetic sequence. The number of flowg r:ru gootcd 39;f7 51,a toashc12scers she is going to plant in the last day of September is 63.

1.Find the common difference of the sequence.
d=  .
2.Find the total number of flowers Gloria is going to plant during September.
$S_{30}$=  .
3.Gloria estimated she would plant 1000 flowers in the month of September. Calculate the percentage error in Gloria's estimate.
ϵ=  %

  The fifth term, $u_5$ , of geometric sequence is 375. The sixth term, $u_6$，of the sequence is 75.
1.Write down the common ratio of the sequence.
r=  .
2.Find $u_1$.
$u_1$=  .
The sum of the first k terms in the sequence is 292968.

3.Find the value of k.
k=  .

  In this question give all answers correct to the4:, :mhebe0u9 vi-nf uf6s ;xc-bcdyi nearest whole 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:
1.1. 10 years;
$u_{10}$=  
1.2. 20 years..
$u_{20}$=  
Find the number of years it will take for the population to reach 15000.
n≈  .

  A 3D printer builds a set of 49 Eiffel Tower Replicas in different s z(8mrey5f v3pizes. The height of the largest tower in this set is 64 cm. The heights of successive smaller towers are 95 % of the p 38 fr5yzpvm(ereceding larger tower, as shown in the diagram below.

1.Find the height of the smallest tower in this set.
$u_{49}$=  .
2.Find the total height if all 49 towers were placed one on top of another.
$S_{49}$=  .

  The second and the t6+.djpi 3 ts7dz f+jlchird terms of a geometric sequence are $u_2$=3 and $u_3$=6.

1.Find the value of r, the common ratio of the sequence.
r=  .
2.Find the value of $u_6$.
$u_6$=  .
3.Find the largest value of n for which $u_n$ is less than $10^4$.
n=  .

  The Australian Koala Foundation estimates that there are about 4 s8i bxxj;0li0t ,cd:b3 cyvhtk:*x9d5000 koalas left in the wild in 2019. A year before, in 2s;v l90,bh:tb:8cck 03x x *tdidixjy018, 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;
2022:=  .
2.the number of years it will take for the koala population to reduce to half of its number in 2018.

  The sum of the first n terms of an1*- nztlv 6tbuldl 5/w arithmetic sequence, $S_n$=$u_1$+$u_2$+$u_3$+...+$u_n$,is given by $S_n$=$2n^2$+n.

1.Write down the values of $S_1$ and $S_2$.
$S_1$=   ; $S_2$=  .
2.Write down the values of $u_1$ and $u_2$.
$u_1$=   ; $u_2$=  .
3.Find d, the common difference of the sequence.
d=  .
4.Find $u_{10}$​ , the tenth term of the sequence.
$u_{10}$=  .
5.Find the greatest value of n, for which ​$u_n$ is less than 100.
n=  .
6.Find the value of n, for which $S_n$ is equal to 1275.
n=  .

  A battalion is arranged, per -r5z;tep0 )azbcqhi -row, according to an arithmetic sequence. There are 24 troops in the third row and 42 troops in the sixth ro5h;tc-airz )eqp0 bz-w.

1.Find the first term and the common difference of this arithmetic sequence.
$u_1$=  .
There are 15 rows in the battalion.

2.Find the total number of troops in the battalion.
$S_{15}$=  troops.

  Charles has a New Years Resolution that he wants to be able to comyis8cuy:mbywx7 5/6(hz7cg cz ;)amwplete 100 pushups in one go without a break. He sets out a training regime whereby he completes 20 pu;)7( cyxwi6h mcm:bz 8y5/ 7awczy ugsshups on the first day, then adds 5 pushups each day thereafter.

1.Write down the number of pushups Charles completes
1.1.on the 4th training day;
$u_4$=  .
1.2.on the nth training day.

On the kth training day Charles completes 100 pushups for the first time.

2.Find the value of k.
k=  .
3.Calculate the total number of pushups Charles completes on the first 10 training days.
$S_{10}$=  .
Charles is also working on improving his long distance swimming in preparation for an Iron Man event in 12 weeks time. He swims a total of 10000 metres in the first week, and plans to increase this by 10 % each week up until the event.

4.Find the distance Charles swims in the 6th week of training.
$v_6$≈  .
5.Calculate the total distance Charles swims until the event.
$S_{12}$≈  m.

  The number of seats in an auditor:obi)b; hxv8 /pns .gqqbl2 e;ium follows a regular pattern where the first row ha:o)bn;vbg;/ x28bqsq.e i lhps $u_1$ seats, and the amount increases by the same amount, d, each row. In the fifth row, there are 62 seats and in the thirteenth row there are 86 seats.

1.Write down an equation, in terms of $u_1$ and d , for the amount of seats
1.1.in the fifth row.
$u_5$=$u_1$+  d

1.2.in the thirteenth row.
$u_{12}$=$u_1$+  d

2.Find the value of $u_1$ and d.
$u_1$=   ; d=   .
3.Calculate the total number of seats if the auditorium has 20 rows.
$S_{20}$=  .

The cost of the ticket for a musical held at the auditorium is inversely proportional to the seat's row. The price for a seat in the first row is 120 dollars, and the price decreases 3% each row. Thus, the value of the ticket for seats in the second row is 116.40 and 112.91 in the third one, etc.

4.1.Find the price of the ticket for a seat in the fifth row, rounding your answer to two decimal places.
p5 =   .

4.2.Find the row of the seat at which the price of a ticket first falls below 70.
n≈  .
4.3.Find the total revenue the auditorium generates by tickets sales if 40 seats in each of the 20 rows are sold. Give your answer rounded to the nearest dollar.
dollar=  .

  Two college students, Dae-u/xq0a9eh veo bscw 3(b+ h-vid and Lisa, decide to invest money they have saved from working part-time jobs. David's investment s+30/ hh(a-weqs- x ev bube9octrategy results in an increase of his investment amount by 1000 each year. Lisa's investment strategy results in her investment amount increasing by 5% each year.

At the start of the second year of investing, David's total investment amount is 21000 and Lisa's is 11655.

1.Calculate
1.1.the original amount David invested.
$u_1$=  .
1.2.the original amount Lisa invested.
$v_1$=  .
During a certain year, n, Lisa's investment amount becomes larger than David's amount for the first time.

2.Find the value of n.
Lisa's amount will become larger than David's during year   .

  Towards the end of 2004, a t/,zw .4n k- vmem*y f)zk.fbpiheatre company upgraded their auditorium and installed new comfortable ergonomic chairs for the audieni.4/y z)-nmbfzk.,m p wfvk* ece.

After the redesign, there were 20 seats in the first row and each subsequent row had three more seats than the previous row.

1.If the auditorium had a total of 16 rows, find
1.1.the total number of seats in the last row.
$u_{16}$=  seats.
1.2.the total number of seats in the auditorium.
$S_{16}$=  seats.
The auditorium reopened for performances at the start of 2005. The average number of visitors per show during that year was 500. In 2006, the average number of visitors per show increased by 5%.

2.Find the average number of visitors per show in 2006.
The average number of visitors per show continued to increase by 5% each year.
we get    visitors
3.Determine the first year in which the total number of visitors to a show exceeded the seating capacity of the auditorium.

The theatre company hosts 25 shows per year.

4.Determine the total number of visitors that attended the auditorium from when it opened in 2005 until the end of 2011. Round your answer correct to the nearest integer.
we get    visitors

  Georgia is on vacation in Costa Rct wzc44 (v3apica. She is in a hot air balloon over a lush jungl34pz(at4c wv ce in Muelle.

When she leans forward to see the treetops, she accidentally drops her purse. The purse falls down a distance of 4 metres during the first second, 12 metres during the next second, 20 metres during the third second and continues in this way. The distances that the purse falls during each second forms an arithmetic sequence.

1.1.Write down the common difference, d, of this arithmetic sequence.
d=  .
1.2.Write down the distance the purse falls during the fourth second.
$u_4$=  m.
2.Calculate the distance the purse falls during the 13th second.
$u_{13}$=  m.
3.Calculate the total distance the purse falls in the first 13 seconds.
$S_{13}$=  m
Georgia drops the purse from a height of 1250 metres above the ground.

4.Calculate the time, to the nearest second, the purse will take to reach
the ground.
n≈   seconds.
Georgia visits a national park in Muelle. It is opened at the start of 2019 and in the first year there were 20000 visitors. The number of people who visit the national park is expected to increase by 8% each year.

5.Calculate the number of people expected to visit the national park in 2020.
$v_2$=  .
6.Calculate the total number of people expected to visit the national park by the end of 2028.
$S_{10}$=  .

  A ball is dropped fromcl7ydz sa )(-s the top of the Eiffel Tower, 324 metres from the ground. Thel y 7cad(s)z-s ball falls a distance of 4.9 metres during the first second, 14.7 metres during the next second, 24.5 metres during the third second, and so on. The distances that the ball falls each second form an arithmetic sequence.

1.1.Find d, the common difference of the sequence.
d=  .
1.2.Find $u_5$ , the fifth term of the sequence.
$u_5$=  m.
2.Find $S_6$, the sum of the first 6 terms of the sequence.
$S_6$=  m.
3.Find the time the ball will take to reach the ground. Give your answer in seconds correct to one decimal place.
we obtain n ≈    seconds
Assuming the ball is dropped another time from a much higher height than of the Eiffel Tower,

4.find the distance the ball travels from the start of the 10th second to the end of the 15th second.
we get S=  m.
The Eiffel Tower in Paris, France was opened in 1889, and 1.9 million visitors ascended it during that first year. The number of people who visited the tower the following year (1890) was 2 million.

5.Calculate the percentage increase in the number of visitors from 1889 to 1890. Give your answer correct to one decimal place.
% increase=  %.
6.Use your answer to part (e) to estimate the number of visitors in 1900, assuming that the number of visitors continues to increase at the same percentage rate.
$v_{12}$=  million.

  The first term of an arithmetic sequence is 24 and the common differen+ 2.gd, wmz)2hujfgdzce is 16. guwzz,ddf 2.2j) h+gm

1.Find the value of the 62 nd term of the sequence.
$u_{62}$=  .
The first term of a geometric sequence is 8. The 4th term of the geometric sequence is equal to the 13th term of the arithmetic sequence given above.

2.Write down an equation using this information.
8$r^x$=y; x=  ; y=  .
3.Calculate the common ratio of the geometric sequence.
r=  .

  Peter is playing on a swing during a school lunch br6;vegn v*:zlbeak. The height of the first swing was 2 m and every subsequent swing was 84 l ng;b*evz:6v% of the previous one. Peter's friend, Ronald, gives him a push whenever the height falls below 1 m.

1.Find the height of the third swing.
we find $u_3$ ≈    m.
2.Find the number of swings before Ronald gives Peter a push.
Hence the swing needs a push after    swings.
3.Calculate the total height of swings if Peter is left to swing until coming
to rest.
$S_∞$=  m.

  Melinda has 300000 in a private foundation. Each year she donates 10% of the y c+pehu1x+t guo7(wx/4*iyq99dul hmoney remaining in her private foundation to cu x+dx9/oiygw7h 1y(*9etlup + 4q hcuharity.

1.Find the maximum number of years Melinda can donate to charity while keeping at least 100000 in the private foundation.
Hence the maximum number of years is   .
Bill invests $400000 in a bank account that pays a nominal interest rate of 4 %, compounded quarterly, for ten years.

2.Calculate the value of Bill's investment at the end of this time. Give your answer correct to the nearest dollar.
FV≈  .

  A bouncy ball is dropped out of a second s:b,mwwx)yw9 ftory classroom window, 5m off the ground. Every time the ball hits the ground it bounces 89% of its previous heigh,yx:9ww) w mbft.

1.Find the height the ball reaches after the 2nd bounce.
$u_2$≈  m.
3.Find the total distance the ball has travelled when it hits the ground for the 5th time.
The total distance travelled by the ball is approximately    metres.

  Landmarks are placed along thaoi/ x7*ebb l/(huem*e road from London to Edinburgh and the distance between each landme/bma(ul/ ho i*xe b7*ark is 16.1 km. The first milestone placed on the road is 124.7 km from London, and the last milestone is near Edinburgh. The length of the road from London to Edinburgh is 667.1 km.

1.Find the distance between the fifth milestone and London.
$u_5$=  .
2.Determine how many milestones there are along the road.
Hence there are    milestones along the road.

  On September 1st, an orchard da(qcl24in8cwct06q0gr3n fir7 f+ ccommences the process of harvesting 36 hectares of apple trees. At the end of September 4th, there were 30 hectares remaining to be harvested, and at the end of September 8th, there were 24 hectares remaining. Assuming that the number of hectares harvested each day is constant, the total number of hectares remaining to be harvested can be de3i6w0c rc+qca(i87 2f f t4nqnrl 0dgcscribed by an arithmetic sequence.

1.Find the number of hectares of apple trees that are harvested each day.
the orchard harvests    hectares of apple trees per day.
2.Determine the number of hectares remaining to be harvested at the end of September 1st.
at the end of September 1st, there are    hectares remaining to be harvested.
3.Determine the date on which the harvest will be complete.
the harvest will be complete at the end of September   th.
​
In 2021 the orchard sold their apple crop for $220000. It is expected that the selling price will then increase by 3.2% annually for the next 7 years.

4.Determine the amount of money the orchard will earn for their crop in 2026. Round your answer to the nearest dollar.
In 2026, the orchard will earn $  .
​
for their crop.
5.1.Find the value of $\sum_{n=1}^8(220000 \times 1.032^{n-1})$. Round your answer to the nearest integer.

5.2.Describe, in context, what the value in part (e) (i) represents.
The total amount earned for the apple crop from    to   
6.Comment on whether it is appropriate to model this situation in terms of a geometric sequence.
the selling price increases at a  ----待选择----similarhigherlower  rate annually.
​

  Consider the sum S=$\sum_{k=4}^l(2k-3)$,where l is a positive integer greater than 4.

1.Write down the first three terms of the series.
  ,  ,  
2.Write down the number of terms in the series.
It is   -  
3.Given that S=725, find the value of l.
l=  

  Let $u_n$ = 5n - 1, for n∈$Z^+$.
1.1.Using sigma notation, write down an expression for $u_1$+$u_2$+$u_3$+...+$u_{10}$.
$u_1$+$u_2$+$u_3$+...+$u_{10}$=$\sum_{k=1}^10(u_k)$,$u_k$=  -1
1.2.Find the value of the sum from part (a) (i).
the value of the sum=  .
A geometric sequence is defined by $v_n$=5×2$^{n-1}$,for n∈$Z^+$.

2.Find the value of the sum of the geometric series $\sum_{k=1}^6(v_k)$.
the sum of 6 terms formula,we get $S_6$=  .

  Let $u_n$ = 5n - 1, for n∈$Z^+$.
1.1.Using sigma notation, write down an expression for $u_1$+$u_2$+$u_3$+...+$u_{10}$.
$u_1$+$u_2$+$u_3$+...+$u_{10}$=$\sum_{k=1}^10(u_k)$,$u_k$=  -1
1.2.Find the value of the sum from part (a) (i).
the value of the sum=  .
A geometric sequence is defined by $v_n$=5×2$^{n-1}$,for n∈$Z^+$.

2.Find the value of the sum of the geometric series $\sum_{k=1}^6(v_k)$.
the sum of 6 terms formula,we get $S_6$=  .

  The sides of a square are 8 cm long.73svg368r. dla u1 w )z0rhko4) de/e mflmaln A new square is formed by joining the midpoints of the r3/z4 lranug17vdlmko3le hfa0de.s8w) )m6adjacent 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.
the total area of the shaded region S≈  cm$^2$
2.Find the total area of the shaded region if the process is repeated indefinitely.
Using the sum of an infinite geometric sequence formula $S_∞$=  cm$^2$

  The half-life, T, in yep.7-k tzanx z1ars, of a radioactive isotope can be modelled by the function akz .zxtp 71n-
$\frac{\ln{0.5}}{\ln{1-\frac{k}{100}}}$ , 0​where k is the decay rate, in percent, per year of the isotope.

1.The decay rate of Hydrogen-3 is 5.5 % per year. Find its half-life.
T(5.5)≈  years
The half-life of Uranium-232 (U-232) is 68.9 years. A sample containing 250 grams of U-232 is obtained and stored as a side product of a nuclear fuel cycle.

2.Find the decay rate per year of U-232.
Solving the equation T(k)=68.9 for k, we obtain k=  %
3.Find the amount of U-232 left in the sample after:

(3.1) 68.9 years;
  grams
(3.2) 100 years.
$u_{101}$≈  grams

  The half-life, T, in days, of a radioactive isotope can bels25r1j x-c hy modelled by the functi5 lxh2-cy1s rjon
$\frac{\ln{0.5}}{\ln{1-\frac{k}{100}}}$,0​
where k is the decay rate, in percent, per day of the isotope.

1.The decay rate of Gold-196 is 6.2 % per day. Find its half-life.
T(6.2)≈  days
The half-life of Phosphorus-32 (P-32) is 14.3 days. A sample containing 120 grams of P-32 is produced and stored in a biochemistry laboratory.

2.Find the decay rate per day of P-32.
Solving the equation T(k)=14.3 for k, we obtain k=  %
3.Find the amount of P-32 left in the sample after:
3.1.
14.3 days;   grams
​
3.2.
50 days. $u_{51}$≈   grams

  Elon is challenged to a spet*9 n. mbbpl ue g4yt42qt l-c-cp-/pied climb at a local mountain. He has to reach a height of 400 metrl- /94t2t4ciug-* .tln q m-pbbppcy ees above the ground within four hours.

Elon knows he can climb 150 metres in the first hour. Due to increasing tiredness, each hour he can only climb 75% of the height climbed in the previous hour.

1.Verify that Elon reaches his target height of 400 metres in four hours.
$S_4$≈  m
The mountain has a height of 650 metres. Elon decides to attempt to climb to the summit.

2.Determine whether he can reach the summit of the mountain if he continues climbing, given his increasing tiredness. Justify your answer.
$S_∞$=  m
On a different day, Elon climbs with energy snacks, which help to reduce his tiredness as he climbs. On this day, Elon again climbs 150 metres in the first hour, but then k% of the height he climbed in the previous hour, where k>75.

3.Calculate the minimum value of k, given that on this day Elon is able to reach the summit. Give your answer as a percentage, to the nearest whole number.
The minimum value of k is    %

  Consider the sequenced3 zd *kkts37kim.td3 $u_1$ , $u_2$ , $u_3$ , ... , $u_n$ , ... where
$u_1$=860, $u_2$=980, $u_3$=1100, $u_4$=1220.
The sequence continues in the same manner.
1.Find the value of $u_{50}$.
$u_{50}$=  .
2.Find the sum of the first 10 terms of the sequence.
$S_{10}$=  .
Now consider the sequence $v_1$ , $v_2$ , $v_3$ , ... , $v_n$ , ... where
$v_1$=2, $v_2$=4, $v_3$=8, $v_4$=16.
This sequence continues in the same manner.
3.Find the exact value of $v_{13}$.
$v_{13}$=  .
4.Find the sum of the first 10 terms of this sequence.
$S_{10}$=  .
k is the smallest value of n for which $v_n$ is greater than $u_n$.

5.Calculate the value of k.k=  .

  On Wednesday Eddy goes to a velj:w)spex(m.k gpv2,t43w rz a odrome to train. He cycles the first lap of the track in 25 ,j4s3 :2 m. xpzprk)t wgavwe(seconds. Each lap Eddy cycles takes him
1. 6 seconds longer than the previous lap.

1.Find the time, in seconds, Eddy takes to cycle his tenth lap.
$u_{10}$=  seconds.

Eddy cycles his last lap in 55.4 seconds.

2.Find how many laps he has cycled on Wednesday.
n=  laps.
3.Find the total time, in minutes, cycled by Eddy on Wednesday.
$S_{20}$=  minutes.
On Friday Eddy brings his friend Mario to train. They both cycled the first lap of the track in 25 seconds. Each lap Mario cycles takes him 1.05 times as long as his previous lap.

4.Find the time, in seconds, Mario takes to cycle his fifth lap.
$v_5$≈  seconds.
5.Find the total time, in minutes, Mario takes to cycle his first ten laps.
$S_{10}$≈  minutes
Each lap Eddy cycles again takes him 1.6 seconds longer that his previous lap.
After a certain number of laps Eddy takes less time per lap than Mario.

6.Find the number of the lap when this happens.
n=  th lap

  A bouncy ball is dropped from a height of 2 metres onto a concrete f)yc87 d 4pupjirrbc3+ /af8drloor. After hittin +ddrcp pbau/f r8)i7yjc438 rg the floor, the ball rebounds back up to 80 % of it's previous height, and this pattern continues on repeatedly, until coming to rest.

1.Show that the total distance travelled by the ball until coming to rest can be expressed by
2+4(0.8)+4(0.8)$^2$+4(0.8)$^3$+...

2.Find an expression for the total distance travelled by the ball, in terms of the number of bounces, n.

3.Find the total distance travelled by the ball.S=  metres.

  A bouncy ball is dropped out of aqk :.lr51)qfgguh m-z second story classroom window, 5 m off the ground. :umzqh51.- )qgkrfglEvery time the ball hits the ground it bounces
89 % of its previous height.

1.Find the height the ball reaches after the 11th bounce.
$u_{11}$≈  m.
2.Find the total distance travelled by the ball until it comes to rest.
total distance travelled≈  m.