“Just because we can’t find a solution it doesn’t mean that there isn’t one.” – Andrew Wiles
Definition
An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference. If a1 is the first term of an arithmetic sequence and d is the common difference, the sequence will be:
{an} = {a1, a1 + d, a1 + 2d, a1 + 3d, …} [1]
To find the nth term of an arithmetic sequence: an = a1 + ((n – 1) * d)
Where:
- an is the nth term in the sequence
- a1 is the first term
- n is the number of the term
- d is the common difference
Who
For anyone who was never introduced to arithmetic or geometric sequences in school (e.g., me) but had to deal with them on the ACT, GMAT and SAT. Sequences are essential not only in math but also computer science and other disciplines.
What
See articles on arithmetic sequences in the Additional Reading section below.
Why
Firstly you need to understand that what is arithmetic progression?
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Difference here means the second minus the first. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with common difference of 2.
Advantages of Arithmetic Progression :-
- Someone can easily determine the future amount.
- We can easily get any far future amount by calculating the nth term of the AP series.
- By calculating nth term we can also get how many instalment we should pay in the payable money amount.
Let’s see where we can apply Arithmetic Progression in real life :-
- Another case is when you are waiting for a transport. Expecting that the traffic is moving at a steady speed you can anticipate the when the following transport (Bus/Cab) will come.
- In the event that you ride a taxi, this additionally has an arithmetic sequence. When you ride a taxi you will be charge an underlying rate and after that a for every mile or per kilometer charge. This show and arithmetic sequence that for each kilometer you will be charge a specific steady rate in addition to the underlying rate.
Arithmetic Progression or Sequence can be connected in all aspects of our lives. We simply need to dissect how it tends to be utilized in our everyday life. Knowing about this sort of sequence can give us an alternate point of view on how things occur in our lives. [2]
See also Theoretical Knowledge Vs Practical Application.
How
I don’t show you how to to compute arithmetic sequences. Many of the References, Additional Reading, websites and videos will assist you with arithmetic sequences.
As some professors say: “It is intuitively obvious to even the most casual observer.”
References
[1] Abramson, Jay. 2014. “Arithmetic Sequences”. openstax. https://opentextbc.ca/precalculusopenstax/chapter/arithmetic-sequences/.
[2] Srivastava, Piyush. “Why Do We Need To Study Arithmetic Progression?”. 2021. Quora. https://www.quora.com/Why-do-we-need-to-study-arithmetic-progression.
Additional Reading
“4 Ways To Find Any Term Of An Arithmetic Sequence – Wikihow”. 2021. wikihow.com. https://www.wikihow.com/Find-Any-Term-of-an-Arithmetic-Sequence.
⭐ “Arithmetic Progression – Definition, nth Term, Sum, and Examples”. 2021. GeeksForGeeks. https://www.geeksforgeeks.org/what-is-arithmetic-progression/?ref=ml_lbp.
“Arithmetic Sequences And Sums”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html.
⭐ “Arithmetic Series – GeeksForGeeks”. 2021. GeeksForGeeks. https://www.geeksforgeeks.org/arithmetic-series/?ref=ml_lbp.
“Lesson: Calculations With Arithmetic Sequences | Nagwa”. 2021. nagwa.com. https://www.nagwa.com/en/lessons/619186850801/.
“Missing Number In A Sequence”. 2021. aaamath.Com. https://www.aaamath.com/patra10.htm.
“Number Sequence Calculator”. 2021. calculator.net. https://www.calculator.net/number-sequence-calculator.html.
kuco. “Proofs Your Teacher Should Have Taught You”. 2022. Medium. https://nseverkar.medium.com/proofs-that-your-teacher-should-have-taught-you-1e3de9be2d87.
In high school, mathematics was done in what I now recognize as a very non-mathematical way. We were told to dogmatically memorize formulas and laws. Because of that, I struggled with many tests and got bad grades. That was until I realized what math truly was. It was not about memorizing stuff, it was about understanding it.
Suddenly everything became simple. Apart from the few definitions, I was able to derive pretty much everything and optimize my brain’s storage space. And later, at the university, this way of studying has turned out to be the right one.
This post is about some easy proofs that most teachers skip. We’re going to clear them up. They’re not difficult but can open a door to studying mathematics the way it was meant to be studied.
McGarry, Mike. 2012. “Sequences On The GMAT”. Magoosh Blog — GMAT® Exam. https://magoosh.com/gmat/sequences-on-the-gmat/.
“Make Your Kid’s Math Teacher Hate You”. 2022. Medium. https://medium.com/mathadam/make-your-kids-math-teacher-hate-you-a258d91dc2d7.
Montgomery, Courtney. 2021. “Sequences On ACT Math: Strategy Guide And Review”. blog.prepscholar.com. https://blog.prepscholar.com/sequences-on-act-math-strategy-guide-and-review.
“Sequences | Algebra 1 | Math | Khan Academy”. 2021. Khan Academy. https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences#x2f8bb11595b61c86:introduction-to-arithmetic-sequences.
“Sequences – Finding A Rule”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/sequences-finding-rule.html.
Videos
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
The featured image on this page is from the nagwa website.
Mathematical Mysteries 