“Just because we can’t find a solution it doesn’t mean that there isn’t one.” – Andrew Wiles

## Definition

### Arithmetic Sequence

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

### Geometric Sequence

A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If a1 is the initial term of a geometric sequence and r is the common ratio, the sequence will be: {a1, a1r, a1r2, a1r3, …}2

## 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 and geometric sequences in the Additional Reading section below.

## Why

### Arithmetic Sequence

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.

1. Someone can easily determine the future amount.
2. We can easily get any far future amount by calculating the nth term of the AP series.
3. 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 :-

1. 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.
2. 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.3

### Geometric Sequence

We encounter geometric sequences in real life, and sometimes we need a formula to help us find a particular number in our sequence. We define our geometric sequence as a series of numbers, where each number is the previous number multiplied by a certain constant. This constant varies for each series but remains constant within one series.

Wondering how this kind of series relates to the real world? You might be surprised to learn that this kind of thing happens all the time, and you might even be involved in it yourself. Imagine that you wanted to spread the message that Friday is the new red scarf day, and everyone should wear a red scarf. First, you send your message to five of your friends. So, we started off with one person, and now we have five new people.

## How

I don’t show you how to to compute arithmetic or geometric sequences. Many of the References, Additional Reading, websites and videos will assist you with sequences. As some professors say: “It is intuitively obvious to even the most casual observer.

1 Abramson, Jay. 2014. “Arithmetic Sequences”. openstax. https://opentextbc.ca/precalculusopenstax/chapter/arithmetic-sequences/.

2 Abramson, Jay. 2014. “Geometric Sequences”. openstax. https://opentextbc.ca/precalculusopenstax/chapter/geometric-sequences/.

3 Srivastava, Piyush. “Why Do We Need To Study Arithmetic Progression?”. 2021. Quora. https://www.quora.com/Why-do-we-need-to-study-arithmetic-progression.

4 “How and Why to Use the General Term of a Geometric Sequence”. 2021. study.com. https://study.com/academy/lesson/how-and-why-to-use-the-general-term-of-a-geometric-sequence.html.

### Arithmetic Sequences

“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 Sequences And Sums”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html.

“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.

### Geometric Sequences

“Geometric Sequence”. 2021. math.net. https://www.math.net/geometric-sequence.

“Geometric Sequences”. 2021. nagwa.com. https://www.nagwa.com/en/videos/493151072621/.

“Geometric Sequences And Sums”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/sequences-sums-geometric.html.

Gregory Grant, and David K. “Why Is A Geometric Progression Called So?” 2015. Mathematics Stack Exchange. https://math.stackexchange.com/questions/1281856/why-is-a-geometric-progression-called-so?noredirect=1&lq=1.

“How To Solve Geometric Sequences”. 2020. effortlessmath.com. https://www.effortlessmath.com/math-topics/geometric-sequences/

“Number Sequence – Explanation & Examples”. 2021. The Story of Mathematics. https://www.storyofmathematics.com/number-sequences.

### Arithmetic & Geometric Sequences

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/.