1.3 Sequences & Series

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4 marks

The sum of the first eight terms of a sequence is 200.

Given that $u_{1}$ = 5.75, find:

(i)

the common difference,

d,

in the case where the sequence is arithmetic

(ii)

the common ratio,

r

, in the case where the sequence is geometric.

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4 marks

Find $S_{12}$ , the sum of the first 12 terms for the arithmetic and geometric sequences found in part (a).

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4 marks

The third term of a geometric sequence is 270 and the sixth term is $-$ 911.25.

Find the 10^th term of the sequence.

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4 marks

Find the sum of the first 21 terms of the sequence.

Give your answer in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in ℤ$ .

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4 marks

The table below shows information about the terms of three different sequences, $a_{n}$ , $b_{n}$ and $c_{n}$ .

	$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 5$	$n = 6$
$a_{n}$	0.1			2.7		24.3
$b_{n}$	24.6		-19		$-$ 62.6
$c_{n}$	880		220			$-$ 27.5

Determine whether $a_{n}$ is an arithmetic or geometric sequence and fill in the table accordingly.

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4 marks

Determine whether $b_{n}$ is an arithmetic or geometric sequence and fill in the table accordingly.

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4 marks

Determine whether $c_{n}$ is an arithmetic or geometric sequence and fill in the table accordingly.

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4 marks

The 18^th term of an arithmetic sequence is 54 and the common difference, $d,$ is 2.2.

Find $S_{18}$ the sum of the first 18 terms of the arithmetic sequence.

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4 marks

The first and 18^th terms of the arithmetic sequence are the first and second terms respectively of a geometric sequence.

Find the smallest value of $n$ such that $S_{n} > 10 000$ for the geometric sequence.

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4 marks

The fifth term of a geometric sequence is 1 and the common ratio, $r$ , is $\frac{1}{3}$ .

Find $S_{5}$ , the sum of the first five terms of the geometric sequence.

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2 marks

Find the exact value of the infinite sequence.

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4 marks

The first and fifth terms of the geometric sequence are the 20^th and 10^th terms respectively of an arithmetic sequence.

Find the largest value of $n$ such that $S_{n} < 1000$ for the arithmetic sequence.

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4 marks

Ashley and Emma are attempting to swim a total of 2000 m each by completing laps of a 25 m pool. Ashley swims her first lap in 17 s and takes 0.2 s longer each lap after that. Emma swims her first lap in 16.5 s and takes 1.01 s times longer each lap after that.

(i)

Find the time Ashley takes to swim her final lap.

(ii)

Find the time Emma takes to swim her final lap.

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4 marks

(i)

State who swims the 2000 m the fastest.

(ii)

Find the mean lap time for both Ashley and Emma.

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1 mark

The table below shows information about the terms of two different sequences, $a_{n}$ and $b_{n} .$

	$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 5$	$n = 6$
$a_{n}$	1		$\frac{9}{4}$			$\frac{243}{32}$
$b_{n}$	$18.6$			$38.1$		$51.1$

State which sequence is arithmetic, and which is geometric.

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4 marks

Fill in the missing values in the table.

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3 marks

Find the largest value of such that $a_{n} < b_{n}$ .

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2 marks

A basketball is dropped from a height of 1 m and bounces on the ground $n$ times. The height that the basketball reaches after each bounce forms a geometric sequence. The height of the basketball after the first bounce is 80 cm and the height after the third bounce is 51.2 cm.

Find the common ratio, $r$ , of the geometric sequence.

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2 marks

Find the height that the ball reaches after the second bounce.

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4 marks

Find the total vertical distance, in metres, travelled by the basketball after the first four bounces.

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2 marks

Find the total distance travelled by the ball.

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4 marks

Since the start of 2020, Malcolm has been on a diet and fitness plan aiming to decrease his waist size. To measure his progress, he has been noting when he goes down a size in trousers. In January he wore a size 46, in April he wore a size 44, in July he wore a size 42 and now, in October, he wears a size 40.

Show that the decrease in Malcolm’s size in trousers forms an arithmetic sequence and find how much his size in trousers decreases each month.

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4 marks

Find the month and year when Malcolm’s size in trousers will be 34.

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1 mark

State a more accurate way Malcolm could measure the reduction in his waist size.

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10a

2 marks

Grace is a photographer and joins Instagram to advertise her photos. She made one post in the first week and four posts in the fifth week.

(i)

Given that the number of posts that Grace makes each week forms an arithmetic sequence, calculate the common difference,

d

(ii)

Comment on the validity of the common difference,

d

, found in part (a).

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10b

2 marks

Find the week in which Grace will make her 1000^thpost.

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10c

2 marks

After 11 weeks Grace has 100 followers and after 21 weeks she has 200 followers.

Assuming the increase in Grace’s followers forms a geometric sequence, calculate:

(i)

r

(ii)

u_{1}

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10d

2 marks

Grace believes that once she reaches 10 000 followers, companies will start paying her to take photographs of their products.

Find the week in which Grace will reach 10 000 followers.

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11a

2 marks

Georgia buys a new computer for herself that costs $1099. At the same time, she buys her son, Duncan, a new gaming computer that costs $2749.

It is anticipated that Georgia's computer will depreciate at a rate of 11% per year, whereas Duncan's gaming computer will depreciate at 18% per year.

Estimate the value of Georgia's computer after 6 years.

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11b

4 marks

Georgia and Duncan's computers will have the same estimated value $k$ years after they were purchased.

Find:

(i)

the value of

k

(ii)

the estimated value after

k

years.

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6 marks

The sixth term of an arithmetic sequence is equal to 3 and the sum of the first 12 terms is 12.

Find the common difference and the first term.

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13a

2 marks

A geometric sequence has $u_{1} = 135$ and $u_{4} = 5 .$

Find the common ratio, $r .$

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13b

2 marks

Find $u_{3} .$

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13c

2 marks

Find $u_{7} .$ Give your answer as a fraction.

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13d

2 marks

Find $S_{\infty} .$

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2 marks

The second term, $u_{2}$ , of a geometric sequence is 44 and the third term, $u_{3}$ , is 55.

Find the common ratio, $r$ , of the sequence.

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2 marks

Find the first term of the sequence, $u_{1}$ .

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2 marks

Find $S_{5}$ , the sum of the first 5 terms of the sequence.

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3 marks

The sum of the first 16 terms of an arithmetic sequence is 920.

Find the common difference, $d$ , of the sequence if the first term is 27.5.

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3 marks

Find the first term of the sequence if the common difference, $d$ , is 11.

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3 marks

The sum of the first 5 terms of a geometric sequence is 461.12.

Find the common ratio, $r$ , of the sequence if the first term is 200, given that $r > 0$ .

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3 marks

Find the first term of the sequence if the common ratio, $r,$ is -2.

Give your answer correct to 2 decimal places.

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2 marks

The table below shows information about the terms of four different sequences $a_{n}, b_{n}, c_{n}$ and $d_{n} .$

	$n = 1$	$n = 2$	$n = 3$	$n = 4$
$a_{n}$		12	30
$b_{n}$		12	30
$c_{n}$	80			10
$d_{n}$	80			10

Calculate $a_{1},$ $a_{4}$ and the common difference, $d,$ given that $a_{n}$ is an arithmetic sequence.

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2 marks

Calculate $b_{1},$ $b_{4}$ and the common ratio, $r,$ given that $b_{n}$ is a geometric sequence.

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2 marks

Calculate $c_{2}, c_{3}$ and the common difference, $d,$ given that $c_{n}$ is an arithmetic sequence.

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2 marks

Calculate $d_{2}, d_{3}$ and the common ratio, $r,$ given that $d_{n}$ is a geometric sequence.

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3 marks

Students are arranged for a graduation photograph in rows which follows an arithmetic sequence. There are 20 students in the fourth row and 44 in the 10^th row.

(i)

Find the common difference,

d,

of the arithmetic sequence.

(ii)

Find the first term of the arithmetic sequence.

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3 marks

Given there are 20 rows of students in the photograph, calculate how many students there are altogether

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2 marks

Marie is an athlete returning to running after an injury and wants to manage the number of kilometres she runs per week. She decides to run 4 km the first week and increase this by 1.5 km each week.

Find the distance Marie ran in the 10^th week.

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3 marks

Find the week in which Marie runs 26.5 km.

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3 marks

Marie’s coach says she can start preparing for her next race once she has run a total of 220 km.

Find the week in which Marie will complete this.

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4 marks

The eighth term, $u_{8}$ , of an arithmetic sequence is 18 and the common difference, $d,$ is 2.

(i)

Find the first term of the arithmetic sequence.

(ii)

Find the value of

u_{17} .

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4 marks

The first and 17^th terms of the arithmetic sequence are the third and fifth terms respectively of a geometric sequence.

(i)

Find the possible values for the common ratio,

r,

of the geometric sequence.

(ii)

Find the first term of the geometric sequence.

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4 marks

In a geometric sequence, $u_{3}$ = 160 and the common ratio, $r,$ is $\frac{1}{4} .$

(i)

Find the first term,

u_{1} .

(ii)

Find

u_{6} .

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2 marks

Find the value of the infinite sum of the sequence.

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4 marks

The first and third terms of the geometric sequence are the seventh and ninth terms respectively of an arithmetic sequence.

(i)

Find the common difference,

d

, of the arithmetic sequence.

(ii)

Find the first term of the arithmetic sequence.

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3 marks

A sequence can be defined by $a_{n} = 32 - 7 n,$ for $n \in ℤ^{+}$ .

Write an expression for $a_{1} + a_{2} + a_{3} + \dots + a_{12}$ using sigma notation and find the value of the sum.

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3 marks

Write an expression for $a_{4} + a_{5} + a_{6} + \dots + a_{15}$ using sigma notation and find the value of the sum.

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10a

3 marks

A sequence can be defined by $g_{n} = 4 \times 3^{n - 1}$ , for $n \in ℤ^{+}$ .

Write an expression for $g_{1} + g_{2} + g_{3} + \dots + g_{10}$ using sigma notation and find the value of the sum.

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10b

3 marks

Write an expression for $g_{8} + g_{9} + g_{1} 0 + \dots + g_{18}$ using sigma notation and find the value of the sum.

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11a

3 marks

The kiwi is a flightless bird and is a national treasure in New Zealand. At the start of 2021 there were approximately 68 000 kiwi left, with the population decreasing by 2% every year.

Find the expected population size of kiwis in 2030 assuming the rate of decrease in kiwi population remains the same.

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11b

3 marks

Find the year in which the population of kiwis falls below 50 000 assuming the rate of decrease in kiwi population remains the same.

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12a

3 marks

Aaron is working on his cycling in preparation for a triathlon event in 10 months. He cycles a total of 240 km in the first month and plans to increase this by 12.5% each month.

Find the distance Aaron cycles in the fifth month of preparation.

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12b

3 marks

Calculate the total distance Aaron cycles until the triathlon.

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13a

2 marks

A geometric sequence has $u_{1}$ = 0.5 and $r$ = 3.

Find

(i)

u_{4}

(ii)

S_{5} .

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13b

4 marks

An arithmetic sequence has the same $u_{4}$ and $S_{5}$ as the geometric sequence above.

Find $u_{1}$ and $d$ for the arithmetic sequence.

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14a

3 marks

Daniel and Jonah have each been given $5000 to save for university.

Daniel invests his money in an account that pays a nominal annual interest rate of 2.24%,compounded quarterly.

Calculate the amount Daniel will have in his account after 8 years.
Give your answer to 2 decimal places.

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14b

3 marks

Jonah wants to invest his money in an account such that his investment will double in 10 years. Assume the account pays a nominal annual interest of $r$ %, compounded half-yearly.

Determine the value of $r$ .

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15a

3 marks

On his 40th birthday, Robert invests $15 000 into a savings account that pays a nominal annual interest rate of 4.78%, compounded monthly.

(i)

Write an expression for the total value of the investment after n years.
Give your answer to 2 decimal places.

(ii)

Find the total amount in the savings account after 3 and 5 years.

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15b

2 marks

Find the age Robert will be when the amount of money in his account will be 1.5 times the initial amount.

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6 marks

The sum of the first two terms of a geometric sequence is 15.3 and the sum of the infinite geometric sequence is 30. Find the positive value of the common ratio, $r$ .

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3 marks

Julie is starting a new web-based subscription business. She sells her subscriptions for $19.50 per month with customers paying at the start of every month. She has 12 customers ready to sign up in the first month. By the fifth month she has 29 customers.

Given that the increase in customers follows an arithmetic sequence, calculate the number of customers Julie will have in the 9^th month

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4 marks

Calculate the revenue Julie’s business will generate by the 17^th month. Give your answer correct to the nearest dollar.

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4 marks

A football team, SME United, have a new stadium with a maximum capacity of 5000 seats, 1500 seats are reserved for the opposition supporters. SME United have 2195 loyal fans who come to all home matches. The manager has predicted that SME United will gain 45 new loyal fans every match who will come to every home match thereafter.

Based on the manager’s prediction, work out how many matches will be played before the number of unreserved seats run out.

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4 marks

A ticket to one of the matches costs $12.

If the manager's prediction for the increase in loyal fans is correct, and if on average half of the tickets reserved for opposition supporters are sold per game, calculate the revenue that SME United will generate from ticket sales in a 30 match season.

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2 marks

Ben and Sam are both cyclists competing in a 22.5 km race at the Herne Hill Velodrome in London, England. One lap of the velodrome is 450 m.

Ben takes a total of 42 minutes to complete the race.

Calculate Ben’s mean lap time in seconds.

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3 marks

Given that each of Ben’s laps took him 1% longer to complete than the previous one, calculate how long it took him (in seconds) to complete his first and last laps.

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3 marks

Sam completes the first lap in 45 seconds and takes 0.2 seconds longer per lap.

Determine who completed the race the first out of Ben and Sam. Justify your answer.

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4 marks

The first three terms of a geometric sequence are $x + 4$ , $6 x$ and $2 x^{2}$ respectively, where $x \in ℝ, x \neq 0$ .

Find $u_{5}$ , the fifth term of the sequence. Give your answer as a fraction.

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4 marks

Find $S_{7},$ the sum of the first seven terms of the sequence.

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2 marks

Lucy is considering two investment strategies.

Strategy A requires an initial deposit of $100. At the start of the second month a deposit of $115 would need to be made, with monthly deposits at the start of each month thereafter that are each $15 more than the deposit in the previous month.

Strategy B requires an initial deposit of $90. At the start of the second month a deposit of $93.60 would need to be made, with monthly deposits at the start of each month thereafter that are each 4% more than the deposit in the previous month.

Write an expression, using sigma notation, to represent the total amount invested after $n$ months in

(i)

Strategy A.

(ii)

Strategy B.

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3 marks

Find which monthly deposit from Strategy A would be the last one that is greater than the corresponding monthly deposit from Strategy B.

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3 marks

Find after which monthly deposit the total amount invested in Strategy B would exceed the total amount invested in Strategy A.

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1 mark

Joshua is the owner of a new restaurant which is open in the evening from Monday to Friday. The restaurant has a maximum capacity of 50 guests per evening. During the restaurant’s first week they had an average of 24.6 guests per evening and the average spend per guest was $57.55.

Calculate the total amount of revenue the restaurant made in the first week. Give your answer correct to 2 decimal places.

How did you do?

3 marks

During the first week Joshua ran a successful marketing campaign and noticed that during the fourth week the restaurant had an average of 33 guests per evening.

Assuming the growth in average guests per evening follows an arithmetic sequence, find the week during which the restaurant will experience capacity issues.

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4 marks

Calculate the total revenue that the restaurant will generate in its first 10 weeks of being open. Give your answer correct to the nearest dollar.

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4 marks

The first term of both an arithmetic and a geometric sequence is 1 and both sequences have the same second term. The 20^th term of the arithmetic sequence is five times the third term of the geometric sequence.

Find the possible values of the second term.

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4 marks

Find the possible values of the 10^th term for each sequence.

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4 marks

The first four terms of an arithmetic sequence are $(x + y), (2 x + 3), (6 y - 1)$ and $(9 y - x)$ respectively.

Find the values of $x$ and $y$ .

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3 marks

Let $S_{n}$ denote the sum of the first $n$ terms of the sequence.

Find the largest value of $n$ such that $S_{n} < 800$ .

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3 marks

Guy starts a new job where his base salary is $ $x$ per year and his salary will increase by $ $y$ every year for $z$ years.

Given that at the end of $z$ years Guy will have earned $367 200 and his salary after $z$ years will be $37 200, show that

$734 400 = z (x + 37 200)$

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2 marks

Additionally, given that after $\frac{z}{4}$ years Guy’s salary will be $26 400, show that

$14 400 = z y$

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4 marks

Find the values of $x, y$ and $z$ .

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10a

2 marks

In an arithmetic sequence $u_{1} = \log_{b} (\frac{x}{y})$ and $u_{2} = \log_{b} (x),$ where $k > 1$ and $x, y > 0 .$

Show that $d = \log_{b} y .$

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10b

6 marks

Let $x = b^{5}$ and $y = b^{7} .$ Find the value of $\sum_{n = 1}^{6} u_{n} .$

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11a

5 marks

The first three terms of a geometric sequence are $2 p + 3, 3$ and $p - 2,$ where $p \in ℤ .$

Show that $p$ satisfies the equation $2 p^{2} - p - 15 = 0 .$

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11b

4 marks

Given that the sequence has an infinite sum, find the value of

(i)

p

(ii)

r .

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11c

2 marks

Find the sum of the sequence.

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DP IB Maths: AA SL

Topic Questions

1.3 Sequences & Series