SEQUENCE AND SERIES NOTES

INTRODUCTION:

A sequence is termed as the representation of a list of objects or numbers in an ordered form, whereas a series is the sum of these sequences. For example, 1, 3, 6, 9, 12,…. is said to be in a sequence because the list is ordered in the form of the numbers divisible by 3. The sequence and series questions are the most important topics for any competitive exam. The sequence and series calculations are categorised in further subdivisions. They are arithmetic sequences and geometric sequences. Each sequence has a definite property and a formula to solve it. Going through several sequences and series notes will help in the calculation of these questions.

WHAT ARE SEQUENCE AND SERIES?

A sequence is termed as the representation of a list of objects or numbers that are present in the form of a pattern. This sequential pattern of numbers could be finite or infinite, and each number or object present in the list is called the term. In contrast, a series is defined as the sum of the sequence. Mathematically, the series or the sum of these patterned sequences are represented by Sigma notation. The sequence and series are some of the basic topics for any competitive exam, as a number of questions are asked based upon this concept. Overall, a sequence can be defined as an ordered representation of a list of numbers in a certain pattern, whereas the series is the sum of these sequences.

TERMINOLOGY USED IN SEQUENCE AND SERIES:

There are several terminologies related to the sequence and series, which differ from each other on the basis of their respective properties. Insight of the category of sequence and series are:

• Arithmetic Sequence: An arithmetic sequence is defined as a sequence in which the difference of each term is constant. It could be easy to understand the arithmetic sequence with the help of an example, i.e., considering a sequence, a1, a2, a3, a4, …, and this particular sequence will be termed as an arithmetic sequence if,a2–a1=a3–a2=a4–a3=an–an-1. Considering an arithmetic sequence a, b and c, which can also be represented as, b-a = c-b. Simplifying this equation we get, 2b=a+c.
• Arithmetic means: Arithmetic means the average of all the numbers present in a sequence, i.e., the sum of all the terms divided by the number of terms present. For example, consider a, b, c as an arithmetic sequence. Arithmetic mean will be represented as, c=(a+b)2
• Geometric Sequence: Geometric Sequence consists of a series of numbers that are distinctive to each other over a common ratio. Therefore, in a geometric sequence, every term other than the first term is the product of the common ratio and the preceding term. Considering a sequence a, b and c, that are in a geometric sequence, can be represented as b2=ac.
• Geometric mean: Geometric mean is defined as a term between the two elements present. For example, consider geometric sequences a, b and c. Here, b is termed as a geometric mean. 
• Harmonic Sequence: All of that sequence or the series of numbers whose reciprocal can represent an arithmetic sequence is termed as the harmonic sequence. For example, (1/5), (1/10), (1/15), (1/20), … can be termed as a harmonic sequence because the reciprocal will give you 5,10,15,20, … which is an arithmetic sequence.

SEQUENCE AND SERIES FORMULA:

Some of the basic Sequence and Series formula are:

• Sum of n terms of an Arithmetic Sequence:

Sn=n2[2a+(n-1)d]=n2(a1+an)

• The formula for the arithmetic means: considering a sequence, a1, a2, a3, a4, …., an. The formula is expressed as,

A=a1+a2+a3+…+ann

• Sum of n terms of the geometric sequence:

Sn=a(1-rn)1-r, if |r|<1

Sn=a(rn-1)r-1, if |r|>1

Sn= {an , if |r|=1}

EXAMPLE:

1. Find the 10th term of the A.P.: 2,4,6, …

a=2 and d=4-2=2

As, tn=a+(n-1)d

t10=2+(10-1)2=2+18=20

The 10th term of the sequence is 20.

1. Is 119 a term of A.P.: 5, 11, 17, …?

Here, a=5, d=11-5=6

tn=119

tn=a+(n-1)

119=s+(n-1)*6

(n-1)=119-5/6

n=20 

CONCLUSION:

The sequence and series questions are generally asked in any of the competitive exams. A brief knowledge about the topic related to sequence and series is quite necessary. A sequence is defined as any set of numbers that are arranged in an ordered pattern, whereas a series is termed the sum of those sequences. The sequence is further classified as arithmetic sequence and geometric sequence that have their own distinctive properties. The sequence and series questions are based upon their respective formula. Going through the sequence and series notes and practising a number of questions will help in understanding the topic better.