Sequence And Series

An itemised collection of elements in which repetitions of any sort is allowed is known as a sequence. They are very similar to sets and the members of the sequence are called as elements (or terms). The length of a sequence is equal to the number of terms and it can be either finite or infinite. The primary difference between a sequence and a set is that in a sequence, individual terms can occur repeatedly in various positions.

Position of a Term in a Sequence

The position of a term in a sequence is its rank or index and denotes its position. There are two conventions for ranking the terms – in one convention, the first term is assigned a rank of zero and in the other convention, the first term is assigned a rank of one. If any term is designated as the nth element in a sequence, then the term is subscripted by n. For example, in a sequence of rational numbers, the nth element may be denoted by Rn. There can be two sequences with exactly the same elements but yet they will be denoted as two different sequences only because the position of the elements change. Consider the anagrams, SCARE and ACRES. If these two are different sequences, the position of S in sequence 1 is 1 and in sequence 2 its position is 5.

Sequence and Series Types and Examples

Some of the most common examples of sequences are:

  • Arithmetic Sequences
  • Geometric Sequences
  • Harmonic Sequences
  • Fibonacci Numbers

Arithmetic Sequence

A sequence in which every term is created by adding or subtraction a definite number to the preceding number is an arithmetic sequence. The definite number added or subtracted throughout the sequence is known as the common difference. Example: Consider the sequence, 4,7,10,13,16,19,22…… The common difference 3 is added to the preceding term to get the next term. The nth term of the arithmetic sequence is denoted by the term Tn and is given by Tn = a + (n-1)d, where “a” is the first term and d, is the common difference. In the example we considered, we can calculate the 21st term as: T21 = 4 + (21-1)3 = 4+60 = 64.

Geometric Sequence

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence. The definite number used to carry out multiplying or dividing operation is known as the common ratio. Example: Consider the sequence 1,4,16,64,256,1024….. The common ratio is 4 and the preceding term is multiplied by 4 to obtain the next term. The nth term of the geometric sequence is denoted by the term Tn and is given by Tn = a (r)n-1 where a is the first term and r is the common ratio. In the example we considered above, the 9th term is can be calculated as T9 = 1 (4)9-1 = 48 = 65536

Harmonic Sequence

A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.

Fibonacci Numbers

Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

Series

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. If a1,a2,a3,a4 …. is a sequence, then the corresponding series is given by

SN = \(\sum_{n=1}^{N}a_{n}\) = a1+a2+a3 + .. + aN

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Practise This Question

Which number would replace the question mark in series
7,12,19,?,39