Sequences & Series (DP IB Analysis & Approaches (AA): HL): Exam Questions

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.

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

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 $r>0$.

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

Give your answer correct to 2 decimal places.

The numbers of students in the rows of a graduation photograph form an arithmetic sequence. There are 20 students in the fourth row and 44 students in the 10th row.

(i) Find the common difference, $d,$ of the arithmetic sequence.

(ii) Find the first term of the arithmetic sequence.

There are 20 rows of students in the photograph.

Find the total number of students in the photograph.

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.

Find the week in which Marie runs 26.5 km.

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.

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 $u_{17}.$

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.

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 $u_6.$

Find the sum to infinity of the sequence.

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.

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.

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

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.

Find the total distance Aaron cycles in the 10 months of preparation.

A geometric sequence has $u_1$= 0.5 and $r$ = 3.

Find

(i) $u_4$

(ii) $S_5.$

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.

In this question, give all answers correct to two decimal places.

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.

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 rate of $r$%, compounded half-yearly.

Find the value of $r$.

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 down an expression for the total value of the investment after $n$ years.

(ii) Find the total amount in the savings account after 3 years and after 5 years. Give your answers correct to 2 decimal places.

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

The sum of the first two terms of a geometric sequence is 15.3 and the sum to infinity of the sequence is 30.

Find the positive value of the common ratio, $r$.

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

Find the 10^th term of the sequence.

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

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

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.

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

The table below shows information about the terms of two different sequences, $a_n$ and $b_n.$

State which sequence is arithmetic, and which is geometric.

Fill in the missing values in the table.

Find the largest value of $n$ such that $a_n<b_n$.

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.

The table below shows information about the terms of three different sequences, $a_n$, $b_n$ and $c_n$.

Determine whether $a_n$ is an arithmetic or geometric sequence and fill in the table accordingly.

Determine whether $b_n$ is an arithmetic or geometric sequence and fill in the table accordingly.

Determine whether $c_n$ is an arithmetic or geometric sequence and fill in the table accordingly.

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.

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

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)(i).

Find the week in which Grace will make her 1000th post.

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) $u_1$

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.

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.

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.

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.

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.

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.

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.

Calculate the total revenue Julie's business will generate in the first 17 months. Give your answer correct to the nearest dollar.

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.

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.

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

Determine who finished the race first. Justify your answer.

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 after each match. These fans 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.

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.

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.

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 $S_n>10000$ for the geometric sequence.

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 each lap after that takes 1.01 times as long as the lap before.

(i) Find the time Ashley takes to swim her final lap.

(ii) Find the time Emma takes to swim her final lap.

(i) Determine who completes the 2000 m in the shorter time.

(ii) Find the mean lap time for both Ashley and Emma.

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.

Find the sum to infinity of the geometric sequence.

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.

Show that the sum of all integers between 600 and 1400 (inclusive) that are not divisible by 7 is equal to $685 885$.

A basketball is dropped from a height of 1 m. 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.

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

Find the total vertical distance, in metres, travelled by the basketball from the point at which it is dropped until it hits the ground for the fifth time.

Find the total distance travelled by the ball.

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

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

$734400=z\left(x+37200\right)$

Additionally, given that Guy's salary in year $\frac{z}{4}$ will be $26 400, show that

$14400=zy$

Find the values of $x,y$ and $z$.

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

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

Find the largest value of $n$ such that $S_n<800$.

In an arithmetic sequence $u_1={\log }_b\left(\frac{x}{y}\right)$ and $u_2={\log }_{b}x$, where $b>1$ and $x,y>0$.

Show that $d={\log }_{b}y$.

Let $x=b^5$ and $y=b^7.$ Find the value of $\sum _{n=1}^6{u}_n.$

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.

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

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

Find $u_5$, the fifth term of the sequence. Give your answer as a fraction.

Find $S_7,$ the sum of the first seven terms of the sequence.

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.

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

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

The first three terms of a geometric sequence with common ratio $r$ are $2p+3$, 3 and $p-2$, where $p\in \mathrm{\mathbb{Z}}$.

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

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

(i) $p$

(ii) $r.$

Find the sum of the sequence.