Sequences and Series
Define sequences and series and understand the classification of sequences (finite and infinite). nth terms of sequences and series
Sequences

An arrangement of numbers in a definite order following some rule is known as a sequence. We also define a sequence as a function whose domain is the set of natural numbers or some subset of the type {1, 2,…, k}.
 For example: 6, 12, 18, 24…; n, n + 1, n + 2, n + 3, n + 4, n + 5; etc.

In general, a sequence is denoted by {a_{n}} or < a_{n}> which represents the sequence a_{1}, a_{2}, a_{3},… a_{n}.
 The numbers a_{1}, a_{2}, a_{3} … and a_{n} occurring in a sequence are called its terms, where the subscript denotes the position of the term.
 The n^{th} term or the general term of a sequence is denoted by a_{n}.

There are two types of sequences: finite and infinite.
 A sequence containing finite number of terms is called a finite sequence. For example: 5, 10, 15, 20, 25, 30, 35 is a finite sequence.
 A sequence containing infinite number of terms is called an infinite sequence. For example: sequence of prime numbers, sequence of natural numbers etc. are infinite sequences.
Note: Sometimes, a sequence is denoted by {T_{n}} or < T_{n}> which represents the sequence T_{1}, T_{2}, T_{3},… T_{n}.
Sum of first n terms of a sequence:
Let {a_{n}} be the sequence such as {a_{n}} = a_{1}, a_{2}, a_{3},… a_{n}. Also, let S_{n} be the sum of its first n terms.
Then we have
S_{n} = a_{1}_{ }+ a_{2}_{ }+ a_{3}_{ }+…+ a_{n}
It can be be observed that:
S_{1} = a_{1}
S_{2} = a_{1}_{ }+ a_{2}
S_{3} = a_{1}_{ }+ a_{2}_{ }+ a_{3}
. . . .
. . . .
. . . .
S_{n} = a_{1}_{ }+ a_{2}_{ }+ a_{3}_{ }+…+ a_{n}
From the above equations, we obtain
S_{1} = a_{1}
S_{2} – S_{1}= a_{2}
S_{3} – S_{2}= a_{3}
. . .
. . .
. . .
S_{n} – S_{n}_{ – 1}= a_{n}
⇒ a_{n }= S_{n} – S_{n}_{ – 1}
So, if S_{n} is known then any term of the sequence can be obtained.
Fibonacci sequence:
If a sequence is generated by a recurrence relation, each number being the sum of the previous two numbers, then it is called a Fibonacci sequence. For example: a_{1} = a_{2} = 1, a_{3} = a_{1}_{ }+ a_{2}, a_{n} = a_{n−2}_{ }+ a_{n−1}, n > 2.
Let's now try and solve the following puzzle to check whether we have understood the concept of Fibonacci sequence.
Series

If a_{1}, a_{2}, a_{3}… a_{n} is a given sequence, then the expression a_{1}_{ }+ a_{2} + a_{3} + …+ a_{n} is called the series associated with the sequence.
 For example:
The series associated with the sequence 18, 36, 54, 72, 90 … is 18 + 36 + 54 + 72 + 90 + …
The series associated with the sequence 2, 4, 6, 8, 10, 12 is 2 + 4 + 6 + 8 + 10 + 12
!  For example:
 
 In a compact form, the series associated with the sequence a_{1}, a_{2}, a_{3}, …, a_{n} can be written in sigma notation as, where sigma ( ∑ ) denotes the sum. !
 
 Note that series a_{1}_{ }+ a_{2} + a_{3} + …+ a_{n} does not refer to the actual sum of the numbers a_{1}, a_{2}, … , a_{n}. Infact, it just refers to the indicated sum, and it shows that a_{1} is the first term, a_{2} is the second term, …, a_{n} is the n^{th} term of the series. !
 
 The series is finite or infinite depending on the given sequence.
For example: The series associated with the sequence of numbers that are multiples of 4 is an infinite series, whereas the series associated with the sequence of numbers that are odd and less than 100 is a finite series.
Solved Examples
Example 1:
The n^{th} term of a sequence is given by a_{n} =. Find the ratio of the 6^{th} term of the sequence to its 4^{th} term.
Solution:
The n^{th} term of a sequence is given by a_{n} =.
Therefore,
Thus, the ratio of the 6^{th} term of the sequence to its 4^{th} term is given by
Hence, the required ratio is 129:17.
Example 2:
Write the first five terms of the sequence whose n^{th} term is given by.
Solution:
It is given that the n^{th} term of the sequence is given by.
Hence, a_{n} =.
On putting n = 1, 2, 3, 4, 5 successively in a_{n}, we obtain
a_{1} =
a_{2} =
a_{3} =
a_{4} =
a_{5} =
Thus, the required first five terms of the sequence are.
Example 3:
Write the first six terms of the series associated with the following sequence:
a_{1} = 5 a…
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