Sequences & Series (DP IB Maths: AI HL)

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

Given that = 5.75, find:

the common difference, in the case where the sequence is arithmetic

the common ratio, , in the case where the sequence is geometric.

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

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 , where  and .

The table below shows information about the terms of three different sequences, ,  and .

	0.1			2.7		24.3
	24.6		-19		62.6
	880		220			27.5

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

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

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

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

Find  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  such that  for the geometric sequence.

The fifth term of a geometric sequence is 1 and the common ratio, , is .

Find , the sum of the first five terms of the geometric sequence.

Find the exact value of the infinite 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  such that  for the arithmetic 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 takes 1.01 s times longer each lap after that.

The table below shows information about the terms of two different sequences,  and 

	1

State which sequence is arithmetic, and which is geometric.

Fill in the missing values in the table.

Find the largest value of such that .

A basketball is dropped from a height of 1 m and bounces on the ground 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, , 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 after the first four bounces.

Find the total distance travelled by the ball.

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.

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

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.

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

Comment on the validity of the common difference, , found in part (a).

Find the week in which Grace will make her 1000^th 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:

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.

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.

A geometric sequence has .

Find the common ratio, .

Find .

Find . Give your answer as a fraction.

Find .

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

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.

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.

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

The first three terms of a geometric sequence are ,  and  respectively, where .

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

Find  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  months in

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.

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.

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.

Let  denote the sum of the first  terms of the sequence.

Find the largest value of  such that .

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

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

Additionally, given that after  years Guy’s salary will be $26 400, show that

Find the values of  and .

In an arithmetic sequence

Show that .

Let and . Find the value of

The first three terms of a geometric sequence are , 3 and , where

Show that  satisfies the equation

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

Find the sum of the sequence.