Sequences and Series
Define sequences and series and understand the classification of sequences (finite and infinite). nth terms of sequences and series
Sequences
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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.
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In general, a sequence is denoted by {an} or < an> which represents the sequence a1, a2, a3,… an.
- The numbers a1, a2, a3 … and an occurring in a sequence are called its terms, where the subscript denotes the position of the term.
- The nth term or the general term of a sequence is denoted by an.
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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 {Tn} or < Tn> which represents the sequence T1, T2, T3,… Tn.
Sum of first n terms of a sequence:
Let {an} be the sequence such as {an} = a1, a2, a3,… an. Also, let Sn be the sum of its first n terms.
Then we have
Sn = a1 + a2 + a3 +…+ an
It can be be observed that:
S1 = a1
S2 = a1 + a2
S3 = a1 + a2 + a3
. . . .
. . . .
. . . .
Sn = a1 + a2 + a3 +…+ an
From the above equations, we obtain
S1 = a1
S2 – S1= a2
S3 – S2= a3
. . .
. . .
. . .
Sn – Sn – 1= an
⇒ an = Sn – Sn – 1
So, if Sn 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: a1 = a2 = 1, a3 = a1 + a2, an = an−2 + an−1, n > 2.
Let's now try and solve the following puzzle to check whether we have understood the concept of Fibonacci sequence.
Series
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If a1, a2, a3… an is a given sequence, then the expression a1 + a2 + a3 + …+ an 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:
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- In a compact form, the series associated with the sequence a1, a2, a3, …, an can be written in sigma notation as, where sigma ( ∑ ) denotes the sum. !--
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- Note that series a1 + a2 + a3 + …+ an does not refer to the actual sum of the numbers a1, a2, … , an. Infact, it just refers to the indicated sum, and it shows that a1 is the first term, a2 is the second term, …, an is the nth term of the series. !--
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- 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 nth term of a sequence is given by an =. Find the ratio of the 6th term of the sequence to its 4th term.
Solution:
The nth term of a sequence is given by an =.
Therefore,
Thus, the ratio of the 6th term of the sequence to its 4th term is given by
Hence, the required ratio is 129:17.
Example 2:
Write the first five terms of the sequence whose nth term is given by.
Solution:
It is given that the nth term of the sequence is given by.
Hence, an =.
On putting n = 1, 2, 3, 4, 5 successively in an, we obtain
a1 =
a2 =
a3 =
a4 =
a5 =
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:
a1 = 5 a…
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