CIE A Level Maths: Pure 1

Topic Questions

4.4 Further Sequences & Series

1a
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2 marks

Lauren is to start training in order to run a marathon.

Each week she will run a number of miles according to the formula

      u subscript n equals 4 n minus 1

where u subscript n is the number of miles to be run in week n .

Work out how far Lauren will run in weeks 1, 2 and 3.

1b
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2 marks

Work out how far Lauren will run in her 10th week of training.

1c
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3 marks

Lauren tends to train for 10 weeks.
Find the total number of miles Lauren will run across all 10 weeks of her training schedule.

 

1d
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1 mark

Explain why the model would become unrealistic for large values of for example for the training schedule of an elite athlete.

 

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2a
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1 mark

Bernie is saving money in order to purchase a new computer.

In the first week of saving Bernie puts £1 into a money box.

In week 2 Bernie adds £2 to the money box, £3 in week 3 and so on.

 

Find the total amount of money in Bernie’s money box after 10 weeks

2b
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2 marks

Show that the total amount of money in Bernie’s money box at the end of week n is £ n over 2 open parentheses n plus 1 close parentheses.

2c
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2 marks

The computer Bernie wishes to buy costs £250.
Will he have saved enough money after 20 weeks?

2d
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1 mark

Give a reason why this might this not be the best way for Bernie to save money?

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3a
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2 marks

The first term of a progression is 4 and the third term is 49.

For the case where the progression is arithmetic, find the second term and common difference of the progression.

3b
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3 marks

For the case where the progression is geometric, find the possible second terms and the corresponding common ratios of the progression.

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4a
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3 marks

A ball is dropped from the top of a building and is allowed to bounce until it comes to rest.  The height the ball reaches after each bounce is modelled by a geometric sequence, with the nth term given by

      u subscript n equals 2 cross times open parentheses 0.8 close parentheses to the power of n minus 1 end exponent

Write down the height the ball reaches after its first bounce and the common ratio between subsequent bounces.

Find the height the ball bounces to after its fifth bounce.

 

4b
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4 marks
(i)
Find the sum of the heights the ball reaches on its first ten 10 bounces.
(ii)
Explain why the total vertical distance travelled by the ball from the moment it
first hits the ground to the moment it returns to the ground after the 10th bounce is twice your answer to part (i).

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5a
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2 marks

Due to soil quality improvements and an expanding business a farmer is able to grow an increasing variety of crops each year according to the formula

      u subscript n equals 3 n minus 1

where u subscript n  is the number of different crops the farmer can grow in year n since they first started the business.

How many different crops was the farmer able to grow in the first year of business?

5b
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2 marks

In which year will the farmer be able to grow exactly 11 different crops?

5c
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2 marks

In which year will the farmer first be able to begin the year growing more than 30 different crops?

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6a
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2 marks

A training track for cyclists is in the shape of a circle and the distance around one lap is 600 m.

A cyclist trains every day for a fortnight, each day increasing the number of laps of the track they complete.  On day 1, they complete 5 laps of the track and increase the number of laps by 3 each day.

Write down a formula for the number of laps, u subscript n, the cyclist completes on day n.

6b
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2 marks

Find the number of laps the cyclist will complete on day 10 of training.

6c
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4 marks
(i)
Find the total number of laps the cyclist will complete over the fortnight.

(ii)
Find the total distance the cyclist will cover over the fortnight.

 

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7a
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4 marks

Two sequences are being used to model the value of a car, nyears after it was new.

At new, the car’s value is £30 000.

 

Model 1 is an arithmetic sequence where the value of the car, u subscript n, at n years old, is given by the formula u subscript n space equals space 30 space 000 space minus space 5000 n.

Model 2 is a geometric sequence where the value of the car, u subscript n, at n years old, is given by the formula u subscript n space equals space 30 space 000 open parentheses 0.6 close parentheses to the power of n.

(i)
Find the age of the car when Model 1 predicts its value has halved.

(ii)
Find the age of the car when Model 2 predicts its value has halved.
7b
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3 marks

Which model predicts the greater value for the car when it is 5 years old?

7c
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3 marks

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1a
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2 marks

Stephen opens a savings account with £600.

Compound interest is paid annually at a rate of 1.2%.

At the start of each new year Stephen pays another £600 into his account.

Show that at the end of two years Stephen has £1221.69 in the account

1b
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2 marks

Show that at the end of year n, the amount of money, in pounds, Stephen will have in his account is given by

      600 left parenthesis 1.012 plus 1.012 squared plus 1.012 cubed plus midline horizontal ellipsis plus 1.012 to the power of n right parenthesis

1c
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2 marks

Hence show that the total amount, in pounds, in Stephen’s account after n years is
      50 space 600 left parenthesis 1.012 to the power of n minus 1 right parenthesis.

1d
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1 mark

State one assumption that has been made about this scenario.

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2a
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2 marks

June is to start training in order to run a marathon.

For the first week of training she will run a total of 3 miles.

Each subsequent week she’ll increase the total number of miles run by x miles.

June intends training for y weeks and will run a total distance of 570 miles during the training period.

Write down an expression in x for the number of miles June will run in the 10th week of training.

2b
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2 marks

Write down an equation in x and y for the total distance June will run during the training period.

2c
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2 marks

Given that June runs twice as far in week 4 than in week 2, find the values of  x and y.

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3a
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4 marks

The first term of a progression is negative 7 x and the second term is x squared.

For the case where the progression is arithmetic with a common difference of negative 6, find the possible values of x and the corresponding values of the fifth term.

3b
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4 marks

For the case where the progression is geometric with a sum to infinity of negative 14, find the third term.

 

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4a
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2 marks

A ball is dropped and bounces such that the height of each bounce is 80% of the height of the previous bounce.

The first bounce of the ball reaches a height of 1.60 space straight m.

Find the height the ball bounces on its 8th bounce.

4b
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2 marks

Find the total vertical distance travelled by the ball from when it first hits the ground until it hits the ground at the end of its twentieth bounce.

4c
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1 mark

Give one reason why this model may be unrealistic.

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5a
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3 marks

A training track for cyclists is in the shape of a circle made up of several lanes

WgkjWI7o_screenshot-2023-08-04-at-8-36-33-am

The shortest, inner lane has a radius of 10 m , with each lane working outwards having a radius 4 space straight m greater than the previous lane.

During a training session a cyclist is expected to complete two laps of each lane, starting with the inner lane, before moving onto the next one. 

You may assume that the lap distance of each lane is the circumference of the circle with the radius indicated above.

Show that the lap distances of each lane form an arithmetic sequence and state the first term and common difference.

5b
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2 marks

Find the distance completed by a cyclist during a training session and using the first six lanes only.

5c
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3 marks

There is a total of 10 lanes on the training track.

If a cyclist wishes to travel a total distance more than twice round each lane, they may continue as many laps around the outside lane as necessary.

A more advanced cyclist wants to travel a distance of at least 6 km during their session. How many laps of the outside lane the cyclist will need to do in total?

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6a
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2 marks

A model maker is constructing part of a model building using a series of hollow tubes, stacked next to each other as illustrated below.

screenshot-2023-08-04-at-8-46-29-am

Each tube is one-tenth shorter than the one to its left.

All tubes are the same width as they are all cut from one longer tube.

Show that no matter how many tubes the model maker uses, the longer tube they are cut from need not be any longer than ten times the height of the tallest tube.

6b
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3 marks

In a different part of the model building, tubes are required to be stacked on top of each other, as shown below.

screenshot-2023-08-04-at-8-50-39-am

The tallest tube is the same length as the tallest tube above.

All the other tubes are 1 space cm shorter than the tube immediately below it.

The longer tube all others are cut from is ten times the height of the tallest tube in the stack.

All of the longer tube shall be used in the stack.

If the length of the tallest tube is and there are tubes in the stack, show that n squared minus left parenthesis 2 a plus 1 right parenthesis n plus 20 a equals 0. 

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1a
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2 marks

Lloyd is to start training in order to run a marathon.

For the first week of training he will run a total of 2 miles.

Each subsequent week he’ll increase the total number of miles run by 3 miles.

He intends training for 15 weeks.

Calculate how far Lloyd will run during his eighth week of training

1b
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2 marks

Work out how much further Lloyd will run in his last week of training compared to his first.

1c
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3 marks

Find the total number of miles Lloyd will run across all 15 weeks of his training schedule.

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2a
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1 mark

Frankie opens a savings account with £400.

Compound interest is paid at an annual rate of 3%.

Show that at the end of the first year Frankie has £412 in the savings account.

2b
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1 mark

At the start of the second year, and each subsequent year, Frankie adds another £400 to the savings account.

Write down the amount of interest the £400 invested at the start of year 2 will earn by the start of year 3.

2c
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2 marks

Explain why the amount of money in the savings account, in pounds, at the end of year 2 can be written as

      left parenthesis 400 cross times 1.03 right parenthesis cross times 1.03 plus 400 cross times 1.03.

2d
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2 marks

Hence show that after years, the amount in pounds in Frankie’s savings account will be

      400 left parenthesis 1.03 plus 1.03 squared plus 1.03 cubed plus midline horizontal ellipsis plus 1.03 to the power of n right parenthesis.

2e
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2 marks

Show that the sum of the geometric series 1.03 plus 1.03 squared plus 1.03 cubed plus midline horizontal ellipsis plus 1.03 to the power of n is given by

      103 over 3 left parenthesis 1.03 to the power of n minus 1 right parenthesis

2f
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2 marks

Hence find the amount of money in Frankie’s savings account at the end of 12 years.

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3a
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4 marks

The first term of a progression is 5 x and the second term is x squared.

For the case where the progression is arithmetic with a common difference of 14

show that x squared minus 5 x minus 14 equals 0

find the possible values of  x and the corresponding values of the third term.

3b
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4 marks

For the case where the progression is geometric with a sum to infinity of 10

explain why x cannot be equal to zero

show that fraction numerator 5 x over denominator 1 minus begin display style x over 5 end style end fraction equals 10

find the third term.

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4a
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2 marks

A ball is dropped, and it bounces to a height of 1.42 space straight m.

Each subsequent bounce reaches a height 60% of the previous bounce.

Show that the heights of bounces form a geometric sequence with first term 1.42 and common ratio 0.6.

4b
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2 marks

Show that the sum of the first 10 terms of this sequence is 3.53 space straight m, to the nearest centimetre.

4c
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2 marks

Hence write down the distance travelled by the ball from when it first hits the ground to when it returns to the ground after its tenth bounce.

4d
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1 mark

A student wants to compare the accuracy of this model with experimental data.
The student decides to investigate what happens on the 25th bounce.
Suggest a problem the student may encounter.

 

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5a
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2 marks

A training track for cyclists is in the shape of a circle made up of several lanes.

 screenshot-2023-08-04-at-8-11-33-am

One lap of the inner lane is 10 straight pi space straight m, with each lane working outwards having a lap distance of 5 straight pi space straight m, more than the lane immediately inside it.

During a training session, a cyclist is expected to complete one lap of each lane, starting with the inner lane, before moving onto the next one.

A cyclist trains until they have completed the first five lanes.
Find the total distance travelled by the cyclist.

5b
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2 marks

There are 8 lanes in total on the training track.

Find the lap distance of the outside lane.

5c
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3 marks

During a particular training session, a cyclist completes a lap of each lane but in addition, also completes a further 10 laps of the outside lane.
Find the total distance the cyclist travels during this training session.

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6a
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2 marks

Two sequences are being used to model the value of a car, n years after it was new.  At new, the car’s value is £25 000. 

Model 1 is an arithmetic sequence.

Model 2 is a geometric sequence.

Both models predict the same value for the car, £7 500, when it is exactly 8 years old.

Find the common difference for Model 1 and the common ratio for Model 2, giving answers to three significant figures where appropriate.

6b
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3 marks

Find the value of the car according to Model 2 in the year Model 1 predicts its value to be £5000.

 

6c
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1 mark

State one benefit of Model 2 over Model 1 for estimating the value of older cars.

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1a
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3 marks

Alex is to start training in order to run a marathon.

For the first week of training Alex will run a total of a miles.

Each subsequent week Alex will increase the total number of miles run by d miles.

Alex intends training for  weeks.

Given that Alex will run 73 miles during the last week of training and a total of 702 miles across the whole training period, show that

      n left parenthesis a plus 73 right parenthesis equals 1404

1b
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2 marks

Given further that during the week halfway through the training schedule
(ie week n over 2) Alex will run 37 miles, show that

      n d equals 72

1c
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4 marks

Find the values of a comma space d space and space n.

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2a
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4 marks

The first term of a progression is x squared and the second term is fraction numerator 6 x over denominator 7 end fraction.

For the case where the progression is arithmetic with a common difference of negative 1 over 7, find the possible values of x and the corresponding values of the eighth term.

2b
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7 marks

For the case where the progression is geometric with a sum to infinity of 7, and given that x space equals space 1 is one of the possible values of  x 

(i)
find all three possible values of x

 

(ii)
determine the first term and common ratio of the corresponding progressions

 

(iii)
show that all three progressions have the same third term and determine the value of that term.

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3a
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4 marks

A ball is dropped from a height of A space straight m.
It bounces to a height of x space m.

Each subsequent bounce height is 25% shorter than the previous bounce.

Show that no matter how many times the ball bounces, it will not travel further than a total distance of open parentheses A space plus space 8 x close parentheses space straight m. 

 

3b
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1 mark

Write down one assumption that has been made using this model when calculating the distance.

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4a
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2 marks

A training track for cyclists is in the shape of a rectangle and two semi-circles as shown below.  The track is also made up of several lanes.

 screenshot-2023-08-04-at-2-41-58-pm

The shortest, inner lane, as shown in the diagram, has straight runs of l space m comma with the semi-circles at each end having a diameter of d space m. 

Each lane moving outwards increases the diameter by  compared to the previous lane.

Show that the distances of each lane form an arithmetic sequence with first term straight pi d space plus space 2 l space straight m
and common difference straight pi e space m

4b
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3 marks

There is a total of 12 lanes on the training track.

Given that l comma space d space and e are integers and that the total distance for all 12 laps is 96 space open parentheses 4 straight pi plus 5 close parentheses space m comma find the value of l and show that 2 d space plus space 11 e space equals space 64.

 

4c
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2 marks

It is recommended that lanes are at least 2 space m wide to allow sufficient space between cyclists in different lanes. 

Find the least value of e and the associated value of d.

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5a
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2 marks

A student is investigating a sequence of values returned from a computer program that has been reported as having a bug in it.  The sequence of values is

      3 comma 1 comma 6 comma space 1 half comma space space 9 comma space space 1 fourth comma space 12 comma space 1 over 8 comma space 15 comma space space 1 over 16 comma space space space...

The bug in the computer program appears to have made it output two different sequences in the same list, rather than outputting them separately.

Suggest what the two sequences could be.

5b
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2 marks

To fix the bug without rewriting or searching the computer program for errors the student decides it would be easier to separate the output into one sequence using the odd numbered terms and another sequence using the even numbered terms.

(i)
Show that the odd numbered terms in the sequence of values form an arithmetic progression, and determine the first term and common difference of the progression.
(ii)
Find an expression for the sum of the first m terms of this progression.
5c
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2 marks

Find an expression for the sum of the first n even numbered terms of the sequence of values.

 

5d
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2 marks

The computer program also outputs the sum of the first 20 terms of the sequence of values. Find the value the computer would output, giving your answer to 4 decimal places.

5e
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1 mark

Does the sum to infinity exist for the sequence of values returned from the computer program? Give a reason for your answer.

 

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6
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5 marks

The first three terms of an arithmetic progression are x space comma space y space and z (in that order).

The first three terms of a geometric progression are also x comma space y and z (in that order).

Determine the mathematical relationship that must exist between x and y for this to be true, and fully describe the two progressions that result.

 

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