The more you study for IIT JEE Mains, the better the chances are that you will score high. Out of all the subjects, Math is one such subject which requires rigorous practice, a deep understanding of every concept, and most importantly, regular study. To help you in your preparation journey and make the ride smoother, Unacademy has come up with a range of chapters and topics which will help you understand each concept in detail without facing any complexities.

Today, in this article, we will be talking about the Sum up to n terms of special series; Sn, Sn1, Sn2, Sn3. As an IIT aspirant, you must have had a brief idea of what special series are or their different types, and also, what arithmetic series is. However, in this article, we will be highlighting these in detail. So, make sure you stick by us till the end and do not skip any parts. So, without any further ado, let’s dive directly into today’s article. 

What does the Series Mean? 

When in a sequence, there are several numbers, the sum of these numbers is termed a series. The series can be both finite and infinite, the same as the sequence. Note, it is given as Sn. 

Let’s understand the series in detail through an easy example- 

Examples of Series

Imagine we have sequence 1, 3, 5, 7, … with us, 

The series of this sequence will mostly be 1 + 3 + 5 + 7 + … 

Which means, we can write it as Sn = 1 + 3 + 5 + 7 + …

Let’s take another example 

Imagine we have a sequence 1, 2, 3, 4, 5, 6,….. With us, 

The series of this sequence will mostly be 1 + 2 + 3 + 4 + 5 + 6 +…. 

Which also means that the sequence can be written as Sn = 1 + 2 + 3 + 4 + 5 + 6 +…. 

Types of Series 

Like every other thing on this planet earth, the series is divided into different types. The following are the types of the series- 

1. Arithmetic series 
2. Geometric series 

Let’s deeply study both of these. Make sure you’re digging deep into the types of series as it’s the most important component of Sum up to n terms of special series; Sn, Sn2, Sn3 when studying for IIT JEE Mains. Without any further ado, let’s discuss the two types of series. 

Arithmetic Series 

The arithmetic series can be defined as the total sum of the terms in an arithmetic sequence, which means the total difference between the preceding term and its successive term is equal each time, then it is known as an arithmetic series.

If you observe the above-mentioned image closely, it is pretty evident that the next term is attained by adding a constant term of 3 every time. Hence, if a = 2, the common difference is 3. We can write the arithmetic series in the following way- 

{a + (a + d) + (a + 2d) + (a + 3d) + ………} 

Here, the first term of the series is given as a, whereas the common difference is given as d. 

The nth term Formula of arithmetic series 

We already know, 

In an arithmetic series, 

the first is given as a 

the difference is donated as d 

the total terms are given as n, 

Hence the formula to calculate the nth term of an arithmetic series will be 

an = a + (n – 1) d 

Sum of Arithmetic Series 

The above given is an arithmetic series with n = 5, a or the first term = 1 and the difference or the d = 1. 

Let’s calculate its sum by using the formula 

Let’s deeply understand the arithmetic series through an easy example! 

Example of the Arithmetic Series 

It is already given that a or the first term of the series = 6, d or the common difference = 2, and n = 5. Find the sum of the first 5 terms in the sequence 

If we put the given values in the formula then, 

With this, let’s jump onto another type of series which is the geometric series- 

Geometric Series 

In simple terms, the sum of the term in a geometric sequence is termed the geometric series. In case, the ratio of every successive term along with its preceding term remains constant at all times, it is said to be a geometric series. 

If we look at the above image closely, here, the next term is obtained by multiplying ½ which is a constant term. Therefore, it can be concluded that the first will surely be ½ and the common ratio given as r will also be the same that is ½. 

nth Term Formula in Geometric Series 

Geometric series is generally written as- 

Where, a is termed to the first term in the series, 

And the common ratio is given as r 

Hence, the nth term formula in geometric series will be 

an = a1 r n – 1 

Here, n is the number of terms  

Sum of Geometric Series 

The sum of the geometric series can be calculated as 

Example of the Geometric Series 

Calculate the sum of the series 

Solution: 

The first step towards finding the sum of the series is determining if the given series is a geometric series or an arithmetic series. It is already given that a = 2, n = 5, and r = 2 

Special Series 

As the name suggests, the special series are special in some or the other way. These series can be both geometric or arithmetic. Let’s look at some of the examples of the special series- 

• 1 + 2 + 3 +… + n
• 12 + 22 + 32 +… + n2
• 13 + 23 + 33 +… + n3

The first equation is the sum of the n natural numbers whereas the second and third given equation is the sum of the squares and cubes of the first n natural numbers. 

Sum of n Terms of the Special Series 

Let’s talk about an example of it! 

Sn = 1 + 2 + 3 + 4 +… + n 

Here, the first step is to find the sum of the natural number given as n  

Looking at the series, it can be determined that its an AP or arithmetic progression since a and d are both equal and 1 

Putting it into the formula of AP, 

Conclusion: 

Having a good hand in maths is truly a blessing when you appear for IIT JEE Mains; no matter how fast you grasp concepts, a thorough understanding and religious practice is a must when you aim higher. Today, in this article, we read about the sum up to n terms of special series; Sn, Sn2, Sn3, along with types of series, arithmetic series, geometric series in detail, which must have given you a deep understanding of what actually works under the hood. 

Over the past ten years, it has been recorded that a decent amount of questions are being asked from this very chapter which means it definitely holds a lot of significance in every student’s life aiming to score higher grades. 

CTA: 

Special series, the sum of it and other related topics that we discussed just in this chapter acts as a base to a lot of upcoming chapters in Math which will help you understand every point easily. In case you face any difficulty while preparing for IIT JEE Mains, visit the Unacademy page to get faster solutions!