## REAL LIFE PROBLEMS INVOLVING ARITHMETIC SERIES

Problem 1 :

A construction company will be penalized each day of delay in construction for bridge. The penalty will be $4000 for the first day and will increase by $10000 for each following day. Based on its budget, the company can afford to pay a maximum of $ 165000 toward penalty. Find the maximum number of days by which the completion of work can be delayed

Let us write the penalty amount paid by the construction company from the first day as sequence

4000,5000,6000,..............

The company can pay 165000 as penalty for this delay at maximum.

So, we have to write this amount as series

4000 + 5000 + 6000 +.....

and the sum of the penalty amount is 165000.

Sn = 165000

(n/2)[2a +(n - 1)d] = 165000

Substitute a = 4000 and d = 1000.

(n/2)[2(4000) + (n -1)1000] = 165000

(n/2)[8000 + (n - 1)1000] = 165000

(n/2)[8000 + 1000n - 1000] = 165000

(n/2)[7000 + 1000n] = 165000

n[7000 + 1000n] = 165000 x 2

7000n + 1000n 2 = 330000

Divide each side by 1000.

7n + n 2 = 330

n 2 + 7n - 330 = 0

(n - 15)(n + 22) = 0

n = 15, -22

Here 'n' represents number of days delayed.

So it must be positive.

Therefore the correct answer is 15.

Problem 2 :

The sum of $1000 is deposited every year at 8% simple interest. Calculate the interest at the end of each year. Do these interest amounts form an A.P?. If so,find the total interest at the end of 30 years.

First let us find the interest using simple interest formula.

Simple Interest = (PNR) / 100

Substitute P = 1000, N = 1 and R = 8.

Simple Interest = (1000 x 1 x 8) / 100

Simple Interest = 80

In the first year, amount deposited is $1000.

Then, interest earned at the end of the first year is

= $80.

In the second year, amount deposited is $1000.

Total deposit for the second year is

= 1000 + 1000

= $2000

= 80 + 80

= $160

So, the interest amounts from the first year are

80, 160, 240..........

This sequence is an arithmetic progression.

Therefore interest amounts form an arithmetic progression.

To find the total interest for 30 years, we have to find the sum of 30 terms in the above arithmetic progression.

Formula to find sum of 'n' terms in an arithmetic progression is

Sn = (n/2) [2a + (n - 1)d]

Substitute a = 80, d = 80 and n = 30.

= (30/2) [2(80) + (30 - 1)80]

= 15[160 + 29(80)]

= 15[160 + 2320]

= 15[2480]

= 37200

So, the total interest earned at the and of 30 years is $37200.

Problem 3 :

If a clock strikes once at 1'o clock,twice at 2'o clock and so on. How many times will it strike a day?

The clock strikes once at 1'o clock, twice at 2'o clock and so on.

1, 2, 3, ..............................12

The above sequences are arithmetic sequences.

Find the sum of terms in both the sequences.

= 2[1 + 2 + 3 + ............ + 12]

Because the sequences are arithmetic progressions, we can use the formula to find sum of 'n' terms of an arithmetic series.

= 2 x (n/2)[a + l]

Substitute n = 12, a = 1 and l = 12.

= 2 x (12/2)[1 + 12]

= 12[13]

= 156

Therefore the clock will strike 156 times in a day.

Problem 4 :

A gardener plans to construct a trapezoidal shaped structure in his garden. The longer side of trapezoid needs to start with a row of 97 bricks. Each row must be decreased by 2 bricks on each end and the construction should stop at 25 th row. How many bricks does he need to buy ?

The longer side of trapezoid shaped garden is containing 97 and each row must be decreased by 2.

The construction has to be stopped when it reaches the 25 th row.

If we write the number of bricks in each row as a sequence, we get

97, 93, 89,............

The above sequence is an arithmetic progression.

We can use the formula to find the sum of 'n' terms of an arithmetic series.

Sn = (n/2)[2a + (n - 1)d]

Substitute a = 97, d = -4 and n = 25.

= (25/2)[2(97) + (25 - 1)(-4)]

= (25/2)[194 + (24)(-4)]

= (25/2)[194 - 96]

= (25/2)(98)

= 25 x 49

= 1225

So, the gardener needs 1225 bricks.

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## 12 Examples Of How Arithmetic Sequence Works In Real-Life

2,4,6,8,10,12, …

Look at these numbers. Did you notice the numbers follow a particular sequence?

If you subtract any two consecutive numbers from the list, you will get a difference of 2.

4 – 2 = 2 6 – 4 = 2 8 – 6 = 2 10 – 8 = 2 12 – 10 = 2

An arithmetic sequence is any set of numbers or integers wherein the difference between any two consecutive numbers remains constant. Hence, we can conclude that the above numbers form an arithmetic sequence.

While you’ll learn this mathematical concept in school through various problems and evaluations related to number sequencing , it is always nice to know how these concepts find their relevance in real life. You will be surprised to know that there are numerous examples around us that we simply don’t notice while we are busy with our mundane tasks. So are you ready to explore some real-life examples that make a beautiful arithmetic sequence? Let’s begin!

## 12 Real-life arithmetic sequences you probably didn’t know about

1. birthdays.

Birthdays are on the same date every year, and with each passing year, you get a year older, not more, not less. This means your birthdays are in an arithmetic sequence because you will get the same difference of one year when you subtract your age in two consecutive years. So if you are 17 this year, you were 16 last year and will be 18 the following year.

## 2. Bank account deposits

To build a stash of savings, many people have the habit of depositing a fixed amount of money in their bank account every month. So, if you deposit, say, $1000 every month into your account, your account will always hold $1000 more than the previous month. It will look something like this: $1000 in January, $2000 in February, $3000 in March, and so on. Do you see how the deposits are forming an arithmetic sequence here?

## 3. Hands of a clock

Did you ever observe the hands of a clock? Each one, whether the seconds, minute, or hour hand, follows a rhythm and moves in an arithmetic sequence. Therefore, the distance moved with each passing second, minute, or hour remains the same throughout the period of 24 hours.

## 4. Stacking chairs

Stackable chairs are designed so you can stack them one above the other to save space during storage. These chairs are another great example of an arithmetic sequence. Try stacking a few chairs. You will observe the height of the stack increases or decreases depending on whether you’re adding or removing a chair from the stack. The difference in height will always remain the same when you study it for two consecutive arrangements.

## 5. Weeks, years, and leap years

Another classic example of arithmetic sequences is weeks and years. No matter which year you consider, it will have a difference of one year from its successor and predecessor. Similar is the case with weeks in a month. Each week follows a cycle, and a new week begins only when seven days of the previous week have passed. Moreover, leap years are also in an arithmetic sequence and have a difference of four years when two successive leap years are subtracted from one another.

## 6. Arrangement of seats in an auditorium

Have you ever been to an auditorium to enjoy audio and visual performances? If yes, did you notice the seating arrangement there? Most auditoriums and open amphitheaters have a continental seating arrangement. This means all seats are arranged in a concave shape facing towards the stage in an arithmetic sequence. Starting from the first row, every following row has an “n” number of seats more than the previous row until the last row, which has the maximum seats in the auditorium.

## 7. Stairs and elevators

Are you one of those who don’t mind climbing a few stairs, or are you someone who waits for the elevator even if you just have to go to the first floor? Well, we are not going to discuss your choice here but tell you that even stairs and elevators are in an arithmetic sequence. Here the constants are the height of each stair and the distance traveled by an elevator between two successive floors.

## 8. Multiples of a number

Say the times table of any number, and you can obtain its multiples. For example,

2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 2 x 4 = 8 2 x 5 = 10

Here, 2,4,6,8, and 10 are the multiples of two. Observe carefully, and you will notice that each multiple is two more than the previous multiple and two less than the next multiple, making the entire set of multiples of 2 an arithmetic sequence. This fact is not just limited to 2 but stands good for multiples of any number, no matter how big or small.

## 9. Seating arrangement

In an event, how many people can you accommodate on a square table? Four, right? That’s one on each side. But if you combine two square tables, how many people can now sit together? Six. Similarly, adding another one to the lot will allow eight people to sit together. You see how the number of people who can sit together is increasing by two people with the addition of each square table. I hope you can appreciate the sequence here.

## 10. An increasing exercise plan

You must be aware that people new to exercising or those who are resuming exercising after a long time are advised to go slow initially and gradually increase the amount of exercise they are performing. This is recommended so that the body gets used to the new routine slowly and builds stamina without the risk of injury. So, if a person starts with one set, he can move to 3 sets the next week, 5 sets the third week, and so on. Here, the exercise plan is an arithmetic sequence the person follows to stay active.

## 11. Decreasing pill dosage

Similar to the above example, when the doctor reduces a patient’s medication dosage gradually and systematically, it forms an arithmetic sequence. For instance, the doctor advises the patient to have seven pills in a week, then moves to 6 pills, then five pills, and continues this pattern until the patient is entirely off the meds.

## 12. Taxi fare

The taxi fare is also an example of an arithmetic sequence. Setting the initial fixed rate aside, the fare increases sequentially for every extra mile traveled. So, if the fixed charge of a taxi is $15 for the first mile, and every extra mile adds $3 to the fixed amount, the sequence of charges formed for five extra miles will be $3, $6, $9, $12, and $15, where the difference is $3 between two consecutive fares.

## Final words

Number sequencing is an essential mathematical concept that is taught to students using various activities and worksheets . An arithmetic sequence is a specific type of number sequence which finds its application in different scenarios. Real-life examples of arithmetic sequences make us realize how math is intricately woven into the world around us. You may not notice it immediately, but a close eye can show you the presence of this interesting mathematical concept in our lives.

By recognizing the patterns in an arithmetic sequence, one can make future predictions, and these predictions can help in a multitude of areas, such as finance, engineering, and more. So, try and seek out new arithmetic sequences in the world around you and see for yourself how this concept applies to your surroundings and everyday lives. You will indeed be left spellbound!

I am Priyanka Sonkushre, a writer and blogger. I am the person behind “ One Loving Mama ,” a mom blog. Equipped with a Bachelor’s degree along with an MBA, my healthcare background helps me deeply understand learning difficulties. I know how challenging it can be for parents to find the right resources to help their children excel in life. So, here I am to blend my healthcare expertise with my parenting experience to create valuable and helpful resources for parents and teachers supporting children with learning differences. If you wish, you can follow me on Facebook and LinkedIn .

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## 8.2: Problem Solving with Arithmetic Sequences

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- Page ID 83159

- Jennifer Freidenreich
- Diablo Valley College

Arithmetic sequences, introduced in Section 8.1, have many applications in mathematics and everyday life. This section explores those applications.

## Example 8.2.1

A water tank develops a leak. Each week, the tank loses \(5\) gallons of water due to the leak. Initially, the tank is full and contains \(1500\) gallons.

- How many gallons are in the tank \(20\) weeks later?
- How many weeks until the tank is half-full?
- How many weeks until the tank is empty?

This problem can be viewed as either a linear function or as an arithmetic sequence. The table of values give us a few clues towards a formula.

The problem allows us to begin the sequence at whatever \(n\)−value we wish. It’s most convenient to begin at \(n = 0\) and set \(a_0 = 1500\).

Therefore, \(a_n = −5n + 1500\)

Since the leak is first noticed in week one, \(20\) weeks after the initial week corresponds with \(n = 20\). Use the formula where \(\textcolor{red}{n = 20}\):

\(a_{20} = −5(\textcolor{red}{20}) + 1500 = −100 + 1500 = 1400\)

Therefore, \(20\) weeks later, the tank contains \(1400\) gallons of water.

- How many weeks until the tank is half-full? A half-full tank would be \(750\) gallons. We need to find \(n\) when \(\textcolor{red}{a_n = 750}\).

\(\begin{array} &750 &= −5n + 1500 &\text{Substitute \(a_n = 750\) into the general term.} \\ 750 − 1500 &= −5n + 1505 − 1500 &\text{Subtract \(1500\) from each side of the equation.} \\ −750 &= −5n &\text{Simplify each side of the equation.} \\ \dfrac{−750}{−5} &= \dfrac{−5n}{−5} &\text{Divide both sides by \(−5\).} \\ 150 &= n & \end{array}\)

Since \(n\) is the week-number, this answer tells us that on week \(150\), the tank is half full. However, most people would better understand the answer if stated in the following way, “The tank is half full after 150 weeks.” This answer sounds more natural and is preferred.

- How many weeks until the tank is empty? The tank is empty when \(a_n = 0\) gallons. Find \(n\) such that \(\textcolor{red}{a_n = 0}\).

\(\begin{array}& 0 &= −5n + 1500 &\text{Substitute \(a_n=0\) into the general term.} \\ 0 − 1500 &= −5n + 1500 − 1500 &\text{Subtract \(1500\) from each side of the equation.} \\ −1500 &= −5n &\text{Simplify.} \\ \dfrac{−1500}{−5} &= \dfrac{−5n}{−5} &\text{Divide both sides by \(−5\).} \\ 300 &= n & \end{array}\)

Since \(n\) is the week-number, this answer tells us that on week \(300\), the tank is empty. However, most people would better understand the answer if stated in the following way, “ The tank is empty after 300 weeks. ” This answer sounds more natural and is preferred.

## Example 8.2.2

Three stages of a pattern are shown below, using matchsticks. Each stage requires a certain number of matchsticks. If we keep up the pattern…

- How many matchsticks are required to make the figure in stage \(34\)?
- What stage would require \(220\) matchsticks?

Let’s create a table of values. Let \(n =\) stage number, and let \(a_n =\) the number of matchsticks used in that stage. Then note the common difference.

Find the value \(a_0\):

\(\begin{array} &a_0 + 3 &= 4 \\ a_0 + 3 − 3 &= 4 − 3 \\ a_0 &= 1 \end{array}\)

The general term of the sequence is:

\(a_n = 3n + 1\)

- Compute \(a_{34}\) to find the number of matchsticks in stage \(34\):

\(a_{34} = 3(\textcolor{red}{34}) + 1 = 103\).

There are \(103\) matchsticks in stage \(34\).

- What stage would require \(220\) matchsticks? We are looking for the stage-number, given the number of matchsticks. Find \(n\) if \(a_n = 220\).

\(\begin{array} &220 &= 3n + 1 \\ 219 &= 3n \\ 73 &= n \end{array}\)

Answer Stage \(73\) would require \(220\) matchsticks.

## Example 8.2.3

Cory buys \(5\) items at the grocery store with prices \(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\) which is an arithmetic sequence. The least expensive item is \($1.89\), while the total cost of the \(5\) items is \($12.95\). What is the cost of each item?

Put the \(5\) items in order of expense: least to most and left to right. Because it is an arithmetic sequence, each item is \(d\) more dollars than the previous item. Each item’s price can be written in terms of the price of the least expensive item, \(a_1\), and \(a_1 = $1.89\).

The diagram above gives \(5\) expressions for the costs of the \(5\) items in terms of \(a_1\) and the common difference is \(d\).

\(\begin{array} &a_1 + a_2 + a_3 + a_4 + a_5 &= 12.95 &\text{Total cost of \(5\) items is \($12.95\).} \\ a_1 + (a_1 + d) + (a_1 + 2d) + (a_1 + 3d) + (a_1 + 4d) &= 12.95 &\text{See diagram for substitutions.} \\ 5s_1 + 10d &= 12.95 &\text{Gather like terms.} \\ 5(1.89) + 10d &= 12.95 &a_1 = 1.89. \\ 9.45 + 10d &= 12.95 &\text{Simplify.} \\ 9.45 + 10d − 9.45 &= 12.95 − 9.45 &\text{Subtract \(9.45\) from each side of equation.} \\ 10d &= 3.50 &\text{Simplify. Then divide both sides by \(10\).} \\ d &= 0.35 &\text{The common difference is \($0.35\).} \end{array}\)

Now that we know the common difference, \(d = $0.35\), we can answer the question.

The price of each item is as follows: \($1.89, $2.24, $2.59, $2.94, $3.29\).

## Try It! (Exercises)

1. ZKonnect cable company requires customers sign a \(2\)-year contract to use their services. The following describes the penalty for breaking contract: Your services are subject to a minimum term agreement of \(24\) months. If the contract is terminated before the end of the \(24\)-month contract, an early termination fee is assessed in the following manner: \($230\) termination fee is assessed if contract is terminated in the first \(30\) days of service. Thereafter, the termination fee decreases by \($10\) per month of contract.

- If Jack enters contract with ZKonnect on April 1 st of \(2021\), but terminates the service on January 10 th of \(2022\), what are Jack’s early termination fees?
- The general term \(a_n\) describes the termination fees for the stated contract. Describe the meaning of the variable \(n\) in the context of this problem. Find the general term \(a_n\).
- Is the early termination fee a finite sequence or an infinite sequence? Explain.
- Find the value of \(a_{13}\) and interpret its meaning in words.

2. A drug company has manufactured \(4\) million doses of a vaccine to date. They promise additional production at a rate of \(1.2\) million doses/month over the next year.

- How many doses of the vaccine, in total, will have been produced after a year?
- The general term \(a_n\) describes the total number of doses of the vaccine produced. Describe the meaning of the variable \(n\) in the context of this problem. Find the general term\(a_n\).
- Find the value of \(a_8\) and interpret its meaning in words.

3. The theater shown at right has \(22\) seats in the first row of the “A Center” section. Each row behind the first row gains two additional seats.

- Let \(a_n = 22 + 2n\), starting with \(n = 0\). Give the first \(10\) values of this sequence.
- Using \(a_n = 22 + 2n\), Find the value of \(a_{10}\) and interpret its meaning in words in the context of this problem. Careful! Does \(n=\) row number?
- How many seats, in total, are in “A Center” section if there are \(12\) rows in the section?

4) Logs are stacked in a pile with \(48\) logs on the bottom row and \(24\) on the top row. Each row decreases by three logs.

- The stack, as described, has how many rows of logs?
- Write the general term \(a_n\) to describe the number of logs in a row in two different ways. Each general term should produce the same sequence, regardless of its starting \(n\)-value.

i. Start with \(n = 0\).

ii. Start with \(n = 1\).

5) The radii of the target circle are an arithmetic sequence. If the area of the innermost circle is \(\pi \text{un}^2\) and the area of the entire target is \(49 \pi \text{un}^2\), what is the area of the blue ring? [The formula for area of a circle is \(A = \pi r^2\)].

6) Three stages of a pattern are shown below, using matchsticks. Each stage adds another triangle and requires a certain number of matchsticks. If we keep up the pattern…

- What stage would require \(325\) matchsticks?

7) Three stages of a pattern are shown below, using matchsticks. Each stage requires a certain number of matchsticks. If we keep up the pattern…

- How many matchsticks are required to make the figure in stage \(22\)?
- What stage would require \(424\) matchsticks?

## Arithmetic Sequence Real Life Problems

A sample document about examples of real life problems about "Arithmetic Sequence" in Mathematics 10

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- 1. SITUATION: SITUATION: There are 125 passengers in the first carriage, 150 passengers in the second carriage and 175 passengers in the third carriage, and so on in an arithmetic sequence.
- 2. PROBLEM: What’s the total number of passengers in the first 7 carriages? SOLUTION: The sequence is 125, 150, 175 … Given: a1= 125; a2= 150; a3= 175 Find: S7=? an = 125+(n-1)25 a7 = 125+(7-1)25=275 We can use the formula: Thus, =1400 Carriage 1st 2nd 3rd … 7th First 7 carriages Number of Passengers 125 150 175 … ? Sn
- 3. SITUATION: SITUATION: There are 130 students in grade one, 210 students in grade two and 290students in grade three in a primary school, and so on in an arithmetic sequence.
- 4. PROBLEM: What’s the total amount of students In the primary school? (Primary School has 6 grades) SOLUTION: The sequence is 130, 210, 290 … Given: a1= 130; a2= 210; a3= 290 Find: S6= ? an = 130+(n-1)80 a6 = 130+(6-1)80=530 We can use the formula: Thus, = 1980 Grade 1st 2nd 3rd … 6th Total from 1st to 6th Grade Number of Students 130 210 290 … ? Sn
- 5. SITUATION: A car travels 300 m the first minute, 420 m the next minute, 540 m the third minute, and so on in an arithmetic sequence.
- 6. PROBLEM: What’s the total distance the car travels in 5 minutes? SOLUTION: The sequence is 300, 420, 540 … Given: a1= 300; a2= 420; a3= 540 Find: S5= ? an = 300+(n-1)120 a5 = 300+(5-1)120=780 We can use the formula: Thus, = 2700 Minute First Second Third Fourth Fifth 5 minutes in Total Distance 300 420 540 … ? Sn
- 7. PROBLEM: SITUATION: A writer wrote 890 words on the first day, 760 words on the second day and 630 words on the third day, and so on in an arithmetic sequence.
- 8. PROBLEM: How many words did the writer write in a week? SOLUTION: The sequence is 890, 760, 630 … Given: a1= 890; a2= 760; a3= 630 Find: s7= ? an = 890-(n-1)130 a7 = 890-(7-1)130=110 We can use the formula: Thus, =3500 Day 1st 2nd 3rd … 7th Whole Week Number of Words 890 760 630 … ? Sn
- 9. SITUATION: You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence.
- 10. PROBLEM: What is the total distance the object will fall in 6 seconds? SOLUTION: Arithmetic sequence: 16, 48, 80, ... Given: a1= 16; a2= 48; a3= 80 Find: S6= ? The 6th term is 176. Now, we are ready to find the sum: Second 1 2 3 4 5 6 Total distance in 6 seconds Distance 16 48 80 … … 176 .....
- 11. SITUATION: The sum of the interior angles of a triangle is 180º,of a quadrilateral is 360º and of a pentagon is 540º.
- 12. PROBLEM: Assuming this pattern continues, find the sum of the interior angles of a dodecagon (12 sides). SOLUTION: Given: d=180 Find: a10= ? This sequence is arithmetic and the common difference is 180. The 12-sided figure will be the 10th term in this sequence. Find the 10th term. 180 360 540 ... ? Sides: 3 4 5 ... 12 Term: 1 2 3 ... ?
- 13. SITUATION: After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6 minutes per day.
- 14. PROBLEM: How many weeks will it be before you are upto jogging 60 minutes per day? SOLUTION: Given: a1 60; d=6 Find: n= ? Adding 6 minutes to the weekly jogging time for each week creates the sequence: 12, 18, 24, ... This sequence is arithmetic. Week Number 1 2 3 … ? Minutes of Jogging each day inside the week 12 18 24 … n
- 15. SITUATION: 20 people live on the first floor of the building, 34 people on the second floor and 48 people on the third floor, and soon in an arithmetic sequence.
- 16. PROBLEM: What’s the total number of people living in the building? SOLUTION: The sequence is 20, 34, 48 … Given: a1= 20; a2= 34; a3= 48 Find: S5= ? Floor 1st 2nd 3rd 4th 5th People living in the building Number of People who live 20 34 48 … ? Sn an = 20+(n-1)14 a5 = 20+(5-1)14=76 We can use the formula: Thus, =240
- 17. SITUATION: Lee earned $240 in the first week, $350in the second week and $460 in the third week, and so on in an arithmetic sequence.
- 18. PROBLEM: How much did he earn in the first 5 weeks? SOLUTION: The sequence is 240, 350, 460 … Given: a1= 240; a2= 350; a3= 460 Find: S5= ? Week 1st 2nd 3rd 4th 5th First 5 weeks Money that Lee Earned $240 $350 $460 … ? Sn an=240+(n-1)110 a5=240+(5-1)110=680 We can use the formula: Thus, =2300
- 19. SITUATION: An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on and has 30 rows of seats
- 20. PROBLEM: How many seats are in the theatre? SOLUTION: Given: a1= 20; a2= 24; a3= 28; n=30 Find: S30= ? Row 1st 2nd 3rd … 30th Total number of rows Number of seats 20 24 28 … ? Sn To find a30 we need the formula for the sequence and then substitute n = 30. The formula for an arithmetic sequence is We already know that is a1 = 20, n = 30, and the common difference, d, is 4. So now we have So we now know that there are 136 seats on the 30th row. We can use this back in our formula for the arithmetic series.

Time Flies Edu

Educational Thoughts, Ideas, and Resources for Teachers

## Examples of Real-Life Arithmetic Sequences

One of my goals as a math teacher is to present real-life math every chance I get. It is not always easy, I have to admit. When I was in college and the earlier part of my teaching career, I was all about the math… not how I might could use it in real life. I’ve made it a goal of mine to find real-life situations. I’ve also tried to catch the situation in action, but it’s not always possible especially since sometimes I think of an idea while driving or when I’m falling asleep at night.

My recent thoughts have been about arithmetic sequences. Seems easy, right? They are linear. There are a ton of linear situations. Yes, but I want visuals! I also did not want the situation to be a direct variation or always positive numbers and always increasing or positive slopes.

Below are some of the situations I’ve come up with along with a picture. I’m happy for you to use these situations with your classes. When you are finished reading this post, please consider filling out this feedback form called: Understanding Our Visitors . Enjoy!

Stacking cups, chairs, bowls etc. (Stacking anything works, but the situations is different when one thing fits inside the other.) The idea is comparing the number of objects to the height of the object.

Pyramid-like patterns, where objects are increasing or decreasing in a constant manner. Ideas for this are seats in a stadium or an auditorium. A situation might be that seats in each row are decreasing by 4 from the previous row. I use this in one of my arithmetic sequence worksheets.

Filling something is another good example. The container can be empty or already have stuff in it. An example could be a sink being filled or a pool being filled. (Draining should also be considered!) The rate at which the object is being filled versus time would be the variables.

Seating around tables. Think about a restaurant. A square table fits 4 people. When two square tables are put together, now 6 people are seated. Put 3 square tables together and now 8 people are seated. I really love this example. You can use a rectangular table as well and start off with 6 seats.

Fencing and perimeter examples are always nice. Discuss how adding a fence panel to each side of a rectangular fence would change the perimeter. Figure one could have one panel on each side (or change it so it isn’t square). Figure two could have two panels on each side. Each time find the new perimeter. The possibilities for fencing are endless. But how fun would it be to get actual toy fence pieces and do this in your classroom?!

Even though this is not particularly a real-life situation, it’s still good because the visual is real life. The students can touch the objects or even create the pattern themselves! Use toothpicks, paperclips or even cereal to make patterns. If you’d rather set them up somewhere in the room for math centers, then that would be good too! The following is an idea with cereal. If you count total Froot Loops, it’s not arithmetic, so it’s best to stick with rows, perimeter, or sides of the triangle to stay with a linear pattern. (Counting all of them is an area problem, so that would make it quadratic.)

Negative number patterns are not as easy to find. Our thoughts usually go to temperature or sea level. There are some fascinating places on earth that are below sea level. I think it would be cool to do a study on some of them. Once you’ve talked about some of these places, then various situations could be created like, during a rainfall the surface of the water started at 215 feet below sea level and rose at a rate of such and such per hour.

Situations involving diving in the ocean could be used as well. Did you know that a diver should descend at a rate no faster than 66 feet per minute or ascend at a rate of no more than 30 feet per minute? I’m sure many students don’t know why and this could certainly create some great accountable talk.

I hope I’ve given you plenty to think about. It’s really fun to create these problems. Students need to know that their math is real and useful! I hope this encourages you to use some of these examples or make up some of your own. I’d love to hear some of your examples. Leave a comment if you’d like. We can all learn from each other!

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Some of the examples I used above are in my Arithmetic Sequence Activity seen below. When I was creating this resource, it really stretched my thinking. I wanted to create something that students could learn from and see how these patterns are involved in real-life situations. I’ve attached a couple more of my resources. I’m working on the geometric sequence activity now and hope to finish in a week or so. The second resource would be a great follow up after teaching arithmetic sequences. It’s a Boom Card Activity. The third resource is an arithmetic and geometric sequence and series game. It is really suited for Algebra 2. The resource at the bottom is a formula chart for geometric and arithmetic sequences and series. It’s a freebie, so take advantage and download from my store!

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## timefliesedu

Math teacher dedicated to sharing teacher tips, ideas and resources. View all posts by timefliesedu

## 5 thoughts on “Examples of Real-Life Arithmetic Sequences”

Thanks a lot ma’am. I really love your examples and it somehow had given me beautiful ideas about arithmetic sequence. It made it clear for me to visualize. Thank you again ma’am❣️

Wow ! U are doing an amazing job …u really helped me to understand better

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Thanks a lot for this amazing examples ma’am it really helps me and my groupmates to perform our group presentation , because of you’re examples and explanation it help us to understand and relate to it more thanksssss❤️❤️❤️❤️

Thanks so much,Mam for these ideas connected to real life.It’s amazing.God bless

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