# Sequences

## Summary

• A sequence is a set of numbers in a given order with a rule for obtaining the terms, written as:   $\left\{ { T }_{ n } \right\} \\$ .
• The nth term of a sequence is an expression that will allow us to calculate the term that is in the nth position of the sequence i.e  ${ T }_{ 1 }$  is the first term of the sequence,  ${ T }_{ 2 }$  is the second term and so on.
• A sequence is called convergent if there is a real number that is the limit of a sequence.
• A recurrence relation for the sequence  $\left\{ { T }_{ n } \right\}$  is an equation that expresses  ${ T }_{ n }$  in terms of one or more of the previous terms of the sequence

A sequence is a set of numbers in a given order with a rule for obtaining the terms. Let’s look at a few set of numbers:

1.       {1, 3, 5, 7…….}
2.       {5, 10, 15, 20……}
3.       {2, 4, 6, 8…..}

The numbers in each of the above set’s are not just written randomly. They all have a certain order. We can find the next number by using a particular rule.

E.g in the above set (1), we can figure out that the next number after 7 would be 9, in set (2) the next number would be 25 and lastly in set (3) the next number would be 10.

Now the question is if we could design a rule for each of these 3 sets to obtain the next number? It is obvious that we can do this as each set follows a pattern. Let’s denote each term in the set by n. the following pattern can be observed in each set.

1.       $n\quad +\quad 2$
2.       $n\quad +\quad 5\quad or\quad 5\quad \times \quad n$
3.       $2\quad \times \quad n$

Hence, such sets are called sequences and each number of the set is called a term of the sequence.

#### What is an n th term?

The nth term of a sequence is an expression that will allow us to calculate the term that is in the nth position of the sequence i.e if we have a set {3, 9, 27, 81…}
The terms in this set are:

$3\quad =\quad { 3 }^{ 1 }$
$9\quad =\quad { 3 }^{ 2 }$
$27\quad =\quad { 3 }^{ 3 }$
$81\quad =\quad { 3 }^{ 4 }$

In this case we can say that the nth term is equal to  ${ 3 }^{ n }$ .

#### Notation

The nth term of any sequence can be written in the notation  ${ T }_{ n }$ , i.e in the set {3, 9, 27, 81…},

${ T }_{ 1 }\quad =\quad 3$
${ T }_{ 2 }\quad =\quad 9$
${ T }_{ 3 }\quad =\quad 27$   and so on…

#### Example #1

Q. Find the 5th term of the sequence { 3, 10, 29, 66….}

Solution:

Since, we have not yet defined any formulas to find the nth term, the basic way is the trial and error method. To find the 5th term of the above sequence, we need to observe the pattern in this sequence.

n = 5 is what we need to find.

We know:

${ T }_{ 1 }\quad =\quad 3$
${ T }_{ 2 }\quad =\quad 10$
${ T }_{ 3 }\quad =\quad 29$
${ T }_{ 4 }\quad =\quad 66$
${ T }_{ 5 }\quad =\quad ?$

 $n$$n$ 1 2 3 4 5 ${ n }^{ 3 }$${ n }^{ 3 }$ 1 8 27 64 125 ${ n }^{ 3 }\quad +\quad 2$${ n }^{ 3 }\quad +\quad 2$ 3 10 29 66 127

This gives us the value of n = 5.

Hence:

${ T }_{ 5 }\quad =\quad 127$         Ans

#### Convergent Sequence

A sequence is called convergent if there is a real number that is the limit of a sequence.

If there is no such number then it is called a divergent sequence. A divergent sequence either grows towards infinity or towards nowhere at all.

Example of a convergent sequence would be set {1000, 100, 10….} as this converges to zero.

#### Recurrence Relations

A recurrence relation for the sequence  $\left\{ { T }_{ n } \right\}$  is an equation that expresses  ${ T }_{ n }$  in terms of one or more of the previous terms of the sequence, namely,  $\left\{ { T }_{ 0 },\quad { T }_{ 1 },\quad { T }_{ 3 }\quad ........\quad { T }_{ n } \right\}$  for all integers n with  $n\quad \ge \quad { n }_{ 0 }$ , where  ${ n }_{ 0 }$  is a non negative integer.

#### Example #2

Q. Let there be a sequence that satisfies the recurrence relation  ${ a }_{ n }\quad =\quad { a }_{ n\quad -\quad 1 }\quad +\quad 3$  for n = 1, 2, 3, 4….  Also let’s suppose that  ${ a }_{ 0 }\quad =\quad 2$.  What are  ${ a }_{ 1 },\quad { a }_{ 2 }\quad and\quad { a }_{ 3 }$?

Solution:

We know that  ${ a }_{ 0 }\quad =\quad 2$  and the given recurrence relation is ${ a }_{ n }\quad =\quad { a }_{ n\quad -\quad 1 }\quad +\quad 3$

Therefore:

${ a }_{ 1 }\quad =\quad { a }_{ 0 }\quad +\quad 3$
${ a }_{ 1 }\quad =\quad { 2\quad +\quad 3\quad =\quad 5 }$

${ a }_{ 2 }\quad =\quad { a }_{ 1 }\quad +\quad 3$
${ a }_{ 2 }\quad =\quad { 5 }\quad +\quad 3=\quad 8 }$

${ a }_{ 3 }\quad =\quad { a }_{ 2 }\quad +\quad 3$
${ a }_{ 3 }\quad =\quad { 8 }\quad +\quad 3\quad =\quad 11$

Hence:

${ a }_{ 1 }\quad =\quad { 5 }$
${ a }_{ 2 }\quad =\quad { 8 }$
${ a }_{ 3 }\quad =\quad { 11 }$      Ans