Introduction to Arithmetic Sequences

Algebra and statistics are members of the same family. Undoubtedly they are tough and complex but we cannot deny their significance in our daily life.

What is a sequence?

Before understanding the arithmetic sequence we will first understand we mean be sequences.

In a classroom, the students sitting represent the sequence they are sitting in the same manner and hence make a sequence. 

They are in order! And each student present in the sequence called “series” let’s put this common concept in a technical example:

We have the list of some numbers which are:

• 4, 3, 4, 65, 7, 6, 4
• 8, 5, 3, 1, 5, 5

What do you think? These numbers in sequence or not? You got it right! These numbers are “not” in the sequence. 

Because in a sequence it is compulsory to be in an “order” just like the students wearing the same uniform, in the same grade, represents the sequence in the classroom. 

So the sequence will be:

• 1, 3, 5, 7, 9…
• 2, 4, 6, 8, 10…

These two lists are in a sequence because each list represents the sequence of “same order” the first row represents the odd numbers and the second shows even numbers. 

The simplest way to predict whether a given list in a sequence or not? In a sequenced series we can always predict the next number. Just like in the above examples we can predict the next even or odd number came in series. 

What is an arithmetic sequence?

Just like a simple sequence, in the arithmetic sequence, we look towards order or the constant value. It is a specific type of sequence in which the difference between two terms is “constant”. We can write the arithmetic sequence as:

{a, a+2d, a+3d,}

In the above expression, we see that “a” is the first term and “d” will be the difference in the terms but remember this difference will be a common difference and this sequence can also be termed as arithmetic progression. 

And the difference between the terms will always remain constant. For example, in a sequence (1, 3, 5, 7, 9,) the difference is two and it is continuous to infinity and the nth term in the series will be:

An = a1 + (n - 1) d

You can use an Arithmetic sequence calculator for quick operation.

What does “limit of function” mean?

You will be surprised to know that the concept of limit is thousand years old but the definition came in the 19th century and the purpose to introduce the “limit” is to show whether a sequence or function approach to a “stable” when its index goes to the fixed point. 

The limit of a function simply means that the given function has a limit at a specific point.

To understand it more clearly let’s consider a function (f) in which f is the real-valued function and b is a continuous quantity. 

 Limit→ bf(x) =L

This limit of the function shows us that the function f(x) can be set close to L if we have arranged the values of x close to b. 

In a given example, x=1, x2-1/x-1 = 12-1/ 1-1 = 0/0

In the above example, we can arrange the value of x very close to 1 and get our desired result. 

We can set it to 0.25 and ultimately the final answer will be: X (2-1) = 1.0625/X-1                                                                                                          

Similarly, if we set the value of x to 0.45 the resultant value will be 1.205, and if we set x=0.9 the function will be 1.810. 

The reason to give multiple examples is to throw light on the fact as we are setting the value closer to 1 the final answer of the function becomes closer to 2. Hence we can say that:

Limit measures the change or the rate of change of a given function by the approximations to get the possible value. 

You can use a limit calculator to calculate it easily, saving your time.  

How are these two concepts used in real life?

If we have talked about the arithmetic sequence, it is used in:

• Filling something…
• Stacking chairs, tables, etc.
• To find, to suggest, or fill any pattern
• Because in every case you will focus on one thing “order”

Similarly, the limit of a function is used:

• Specifically by engineers to design the engines of cars.
• To make calculations by approximating a function by applying small differences and calculate by having smaller spacing in the functions.