Special Sequences and Series

One of the most fundamental topics in Maths is sequence and series. A sequence is a list of items/objects arranged in a sequential way. A series can be highly generalised as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence. Arithmetic Progression is one of the most common instances of sequence and series.

In essence, a sequence is a collection of items or objects that have been arranged in a specific order. The sum of all the terms in a sequence can generalise a series. However, there must be a clear relationship between all the sequence’s terms.

By answering problems based on formulas, the principles of this topic can be better grasped. Sequences and series are similar to sets, but the main difference is that individual terms in a sequence can appear multiple times in different positions. 

A sequence’s length is equal to the number of terms, and it might be finite or infinite. 

Types of Sequence and Series

Here are some examples of different types of sequences:

• Arithmetic sequences

• Harmonic sequences

• Geometric sequences

• Fibonacci numbers

1. Arithmetic Sequences

An arithmetic sequence is a sequence in which each term is formed by adding or subtracting a defined number from the previous number. A sequence a1, a2, a3,…, an an is called an arithmetic sequence/arithmetic progression if each progressive number is a sum or difference of the previous number and a constant. 

The arithmetic series can be written in the form of

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

Formula of the nth term of the Arithmetic Series:

The formula for the nth term will be

an = a + (n – 1) d

,where  a – first term,

d – common difference and

n –  number of the terms

2. Geometric Sequences

A geometric sequence is one in which each term is obtained by multiplying or dividing a defined number by the previous number.

The explicit formula for a geometric sequence is an an = a1rn-1, where r is the common ratio. A geometric sequence can be defined recursively by the formulas a1 = c, an+1 = ran, where c can be defined as the constant while r, on the other hand, is the common ratio.

3. Harmonic Sequences

If the reciprocals of all the elements in a sequence form an arithmetic sequence, it is said to be in a harmonic sequence.

4. Fibonacci Numbers

Fibonacci numbers are a fascinating number series in which each element is created by adding two preceding elements, with the sequence beginning with 0 and 1. 

Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

Sequence and Series Examples

Question 1: Find the sum of first n terms of the series 1.3 + 3.5 + 5.7 + …

Solution:

Let Sn = 1.3 + 3.5 + 5.7 + …

The nth term of the series t n = {nth term of 1, 3, 5, …} × {nth term of 3, 5, 7,…}

= (2n – 1) (2n + 1) = 4n2 – 1 

Sn= ∑ (4n2 – 1)

= 4∑n2 – ∑1

= 4n(n+1)(2n+1)6 – n

= 2n(n+1)(2n+1)3 – n

Question 2: Consider the numbers 1, 4, 16, 64, 256, 1024, and so on… Find the 9th term and the common ratio.

The given series is GP so

4/1 = 4 is the common ratio (r).

To get the following term, multiply the previous term by four.

Tn denotes the nth term of the geometric sequence, which is given by Tn = arn-1

The first term is a, and the common ratio is r.

Here, a = 1, r = 4, and n = 9 are the values.

As a result, T9 = 1* (4)(9-1) = 48 = 65536 can be calculated.

It can be a lot of fun to learn about mathematical principles.

Question 3: 2, 3, 6, n, 42… Find the value of n in the above series.

1. 9

2. 12

3. 15

4. 18

5. 21

The right answer is C. 

2, 3, 6, n, 42… are the numbers in the given series. By multiplying the preceding number by 3 and then subtracting 3 from it, we can decode the series. As can be seen, the first two digits in the sequence are 2.

Conclusion 

Hopefully, you now understand the special sequence and series, arithmetic sequence, types, examples, and more.

The total of all the numbers in a sequence can be defined as a series. The sequences are both finite and endless. Similarly, the series might be either finite or infinite.The concept of these sequences are very important for later chapters so students should complete it thouroughly.