Sequences and Series

Basic Definitions of Sequences and Series

In Number sense, A set of numbers arranged in a definite order according to some definite rule is called a sequenceSequences have wide applications.

For example, the amount of money in a fixed deposit in a bank, over a number of years increases in sequence.

Sequence

A sequence is a function whose domain is the set N of natural numbers.

Or

A set of numbers arranged in a definite order according to some definite rule is called a sequence.

It is customary to denote a sequence by a letter 'a' and the image a(n) or t(n), n $in$ N, under ‘a’ by an or tn.

Examples:

1, 3, 5, 7…..... (Adding 2 to every term)

1, 4, 16, 64 … (Multiplying each term by 4)

20, 17, 14 …. (Add -3 to every term)

The different numbers in a geometric sequence are called terms of the sequence. These terms are usually denoted by a1, a2, a3…an or t1, t2, t3…tn.

The subscripts denote the position of the term.

In the second example, 4 is the second term, and 14 is the third term in the third example.

The nth term of a sequence is called the general term of the sequence and is usually denoted by an or tn.

Infinite Sequences and Series

In Sequences and Series , A arithmetic sequence is called finite if the number of terms is finite. A finite sequence always has a last term.

Examples :

2, 5, 8, 11, 14 …, 32

37, 33 …, 1

A sequence is called infinite if the number of terms is infinite. An infinite sequence has no last term. In this sequence, every term is followed by a new term.

Example 1: A sequence of multiples of 5

5, 10, 15, 20…..

Example 2: A sequence of reciprocals of positive integers

1, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, ........

The above two sequences are clearly infinite sequences. We cover more Problems on Sequence and series, which helps us to understand the concept better.

Progressions

In Sequences and Series , A Sequence is said to be a progression if the absolute value of the terms increases (or decreases).

Series

In Sequences and Series, Indicated sum of the terms in a sequence is called a geometric series . The result of performing the additions is the sum of the series.

The indicated sum a1+a2+a3+…+an of the terms of a sequence a1, a2, a3,…an is called a geometric series.

Example : 1 + 4 + 7 + 10 + ... is a series

Here the first term is 1, second term is 4, third term is 7 and so on.

Integer Sequences

Integer Sequences are a kind of sequences.It is explicity given by its `n` th term and implicity given by the relationship of its terms.

For any given integers a[0] and a [1] , an infinite sequence can be given as:

a [ k + 2 ] = ( 2k + 3)  a[k + 1] +   (k + 1)2  a[k]

There are different examples of Integer sequences like the Fibonacci sequences, Kolakoski sequences etc,.

Fibonacci Sequence

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