# Finding the terms of an arithmetic sequence

Rahul applied for a job and got selected. He is offered a starting monthly salary of $20000, with an annual increment of$500. His salary (in $) for the 1st, 2nd, 3rd,â€¦ years will be 20000, 20500, 21000,â€¦.. In a school, five cash rewards are given to the students for their academic performance. If the first prize is$500 and the subsequent rewards are $50 less than the preceding reward, the value of each reward (in$) will be 500, 450, 400, 350, 300.

Don't use plagiarized sources. Get Your Custom Essay on
Finding the terms of an arithmetic sequence
Just from $10/Page In a fixed deposit scheme, the amount becomes \frac{5}{4} 4 5 â€‹ times itself after every 2 years, then the maturity amount of an investment of$8000 after 2, 4, 6 and 8 years (in $) will be: 10000, 12500, 15625, 19531.25 (in$) .

In the above examples we have seen some pattern being followed. We can see that succeeding terms are obtained by adding with a fixed number, subtracting with a fixed number, multiplying with a fixed number, and so on.

Now we will see how to find nth terms of such arithmetic series.

Common difference of arithmetic sequence:
The fixed number that is added to preceding term to obtain the next term is called the common difference of arithmetic sequence. It can be positive, negative and even zero.

If we denote first term of an AP by {a_1}a
1
â€‹
, second term by {a_2}a
2
â€‹
and so on and nth term by {a_n}a
n
â€‹
and the common difference by d, then the AP becomes {a_1},\,{a_2},\,{a_3}, \ldots ,{a_n}a
1
â€‹
,a
2
â€‹
,a
3
â€‹
,â€¦,a
n
â€‹
.

Therefore, {a_2} – {a_1} = {a_3} – {a_2} = \cdots = {a_n} – {a_{n – 1}} = da
2
â€‹
âˆ’a
1
â€‹
=a
3
â€‹
âˆ’a
2
â€‹
=â‹¯=a
n
â€‹
âˆ’a
nâˆ’1
â€‹
=d.

The general form of arithmetic sequence can be written as a,\,a + d,\,a + 2d,a + 3d,…..a,a+d,a+2d,a+3d,….. where a represents first term of arithmetic sequence and d denotes the common difference.

If we know the first term and the common difference, we can find the arithmetic sequence, because we know that in arithmetic sequence, every succeeding term is obtained by adding d to the preceding term. We can also find d of arithmetic sequence by subtracting any term from its succeeding term.

nth term of an arithmetic sequence:
Let us consider the situation where Rahul applied for a job and got selected. He was offered a starting monthly salary of $20000, with an annual increment of$500. Let us check what will be his monthly salary in the fifth year:

Let us first see his monthly salary for the second year,

{\rm{Salary\ for\ second\ year}} = \left( {20000{\rm{ }} + {\rm{ }}500} \right)Salary for second year=(20000+500)

{\rm{Salary\ for\ third\ year}} = \;\left( {20000 + 500 + 500} \right)Salary for third year=(20000+500+500)
= \left( {20000 + 2 \times 500} \right)=(20000+2Ã—500)
= \left[ {20000 + \left( {3 – 1} \right) \times 500} \right]=[20000+(3âˆ’1)Ã—500]
= 21000=21000

{\rm{Salary\ for\ fourth\ year}} = \left( {21000 + 500} \right)Salary for fourth year=(21000+500)
= \left( {20000 + 500 + 500 + 500} \right)=(20000+500+500+500)
= \left( {20000 + 3 \times 500} \right)=(20000+3Ã—500)
= \left[ {20000 + (4 – 1) \times 500} \right]=[20000+(4âˆ’1)Ã—500]
= 21500=21500

{\rm{Salary\ for\ fifth\ year}} = \left( {21500 + 500} \right)Salary for fifth year=(21500+500)
= \left( {20000 + 500 + 500 + 500 + 500} \right)=(20000+500+500+500+500)
= \left( {20000 + 4 \times 500} \right)=(20000+4Ã—500)
= \left[ {20000 + (5 – 1) \times 500} \right]=[20000+(5âˆ’1)Ã—500]
= 22000=22000

Observe we are getting a list of numbers,

20000, 20500,21000, 21500, 22000,â€¦â€¦

Now looking at this pattern, we can find his salary for any year. Let us try to find for the 15th year:

{\rm{Salary\ for\ 1}}{{\rm{5}}^{th}}{\rm{ year}} = {\rm{Salary}}\,{\rm{for}}\,{\rm{1}}{{\rm{4}}^{{\rm{th}}}}\,{\rm{year}}\,{\rm{ + }}\,{\rm{500}}Salary for 15
th
year=Salaryfor14
th
year+500
{\rm{ = }}\left( {20000 + 13 \times 500} \right)\, + \,500=(20000+13Ã—500)+500
= \left( {20000 + 14 \times 500} \right)=(20000+14Ã—500)
= \left[ {20000 + (15 – 1) \times 500} \right]=[20000+(15âˆ’1)Ã—500]
= 27000=27000

Which is First salary + (15-1) X Annual increment.

This example gives us some idea to write nth term.

Let {a_1},\,{a_2},\,{a_3},…….,{a_n}a
1
â€‹
,a
2
â€‹
,a
3
â€‹
,…….,a
n
â€‹
be an arithmetic sequence whose first term {a_1}a
1
â€‹
is a and the common difference is d.

Then,

Second term is {a_2} = a + d = a + \left( {2 – 1} \right)da
2
â€‹
=a+d=a+(2âˆ’1)d

Third term is {a_3} = {a_2} + d = \left( {a + d} \right) + d = a + 2d = a + \left( {3 – 1} \right)da
3
â€‹
=a
2
â€‹
+d=(a+d)+d=a+2d=a+(3âˆ’1)d
Fourth term is {a_4} = {a_3} + d = \left( {a + 2d} \right) + d = a + 3d = a + \left( {4 – 1} \right)da
4
â€‹
=a
3
â€‹
+d=(a+2d)+d=a+3d=a+(4âˆ’1)d
Looking at this pattern, we can say that nth term {a_n} = a + \left( {n – 1} \right)da
n
â€‹
=a+(nâˆ’1)d where a is the first term and d is the common difference of the arithmetic sequence.

{a_n}a
n
â€‹
is called the general term of the AP. Sometimes, it is also denoted by ll.

Applications of arithmetic sequence:
We use arithmetic sequence very often in our real life. When we ride a taxi, this also has an arithmetic sequence. When we take a taxi, we are charged an initial fixed rate and then a per kilometer charge is applied. This is an arithmetic sequence where for every kilometer you will be charged a certain constant rate plus the initial rate.

We use it in calculating our financing estimates, like simple and compound interest, loan amortization. Arithmetic sequence can be applied in almost all aspects of our lives. We just have to analyze how we use it in our day-to-day life. Having knowledge about this kind of sequence can give us a different perspective on how things happen in our lives.

## Get Professional Assignment Help Cheaply

Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?

Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.

## Why Choose Our Academic Writing Service?

• Plagiarism free papers
• Timely delivery
• Skilled, Experienced Native English Writers
• Ability to tackle bulk assignments
• Reasonable prices

## Online Academic Help With Different Subjects

### Literature

Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.

### Finance

Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.

### Computer science

Computer science is a tough subject. Fortunately, our computer science experts are up to the match. No need to stress and have sleepless nights. Our academic writers will tackle all your computer science assignments and deliver them on time. Let us handle all your python, java, ruby, JavaScript, php , C+ assignments!

### Psychology

While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.

### Engineering

Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.

### Nursing

In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.

### Sociology

Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.

We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!

### Statistics

We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.

### Law

Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.

## What discipline/subjects do you deal in?

We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.

## Are your writers competent enough to handle my paper?

Our essay writers are graduates with bachelor's, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.

## What if I don’t like the paper?

There is a very low likelihood that you won’t like the paper.

### Reasons being:

• When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
• We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.

### In the event that you don’t like your paper:

• The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
• We will have a different writer write the paper from scratch.
• Last resort, if the above does not work, we will refund your money.

## Will the professor find out I didn’t write the paper myself?

Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.

## What if the paper is plagiarized?

We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.

## When will I get my paper?

You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.

We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. Noone will ever know that you used our assignment help services. It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.

## How our Assignment Help Service Works

#### 1. Place an order

You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.

#### 2. Pay for the order

Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.

#### 3. Track the progress

You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.

The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.

GET A PERFECT SCORE!!!

## Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
\$26
The price is based on these factors:
Number of pages
Urgency
Basic features
• Free title page and bibliography
• Unlimited revisions
• Plagiarism-free guarantee
• Money-back guarantee
On-demand options
• Writer’s samples
• Part-by-part delivery
• Overnight delivery
• Copies of used sources
Paper format
• 275 words per page
• 12 pt Arial/Times New Roman
• Double line spacing
• Any citation style (APA, MLA, Chicago/Turabian, Harvard)

# Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

### Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

### Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

### Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.