A sequence of events

A series of Mathematics tests

Cricket test match Series.

Succession of events

In Mathematics

"Sequence",

"Series"

have different technical meaning.

List of even integers

2,4,6,8, ...., 2n, .....

Dividing 10 by 3 successive quotient

• Obtained when we divide 10 by 3 at different step,
• 3, 3.3, 3.33, .....

"Rabit problem".

• How many pairs of rabits are there at the end of one year?

Fibonacci numbers

1 $\rightarrow$ Start

1 $\rightarrow$ at the end of 1 month

2 $\rightarrow$ at the end of $2^{nd}$ month

3 $\rightarrow$ at the end of three months

5 $\rightarrow$ at the end of 4 months

Bank pays interest at the rate of 10% per annum.

Invests Rs. 1 at the bank.

Amount = Principal + Interest

I = P N R

$=1 \times 1 \times \frac{1}{10}=\frac{1}{10}$

Amount = 1 + $\frac{1}{10}$.

$1+1 \times \frac{1}{2} \times \frac{1}{10}=1+\frac{1}{20}$

at the end of first half year.

$(1+\frac{1}{20}) \cdot \frac{1}{2} \cdot \frac{1}{10} =(1+\frac{1}{20}) \frac{1}{20}$

Total amount $=(1+\frac{1}{20})+\frac{1}{20}(1+\frac{1}{20})$

$=(1+\frac{1}{20})^2$

Amount at the end of 1 year (interest is calculated compounded fashion, thrice an year)

$=(1+\frac{1}{30})^3$

If bank "compounds" interest n times a year

Amount after one year $=(1+\frac{1}{10 n})^n.$

$1+\frac{1}{10},(1+\frac{1}{20})^2,(1+\frac{1}{30})^3$

$\cdots \cdots (1+\frac{1}{10 n})^n.$

"ordered list of numbers"

Ordered list of numbers.

eg: 1, 3, 5, 7, 9 ......

$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$ ......

$a_1, a_2, a_3, \ldots a_n$, .......

$a_i$'s are numbers.

$1 \rightarrow a_1$

$2 \rightarrow a_2$

$3 \rightarrow a_3\ldots\ldots$

A (real) sequence is a function "f" from set of natural numbers $\mathbb{N}$ to the set of $\mathbb{R}$ reals.

$f: \mathbb{N} \rightarrow \mathbb{R}$

$f(n)=a_n$

2, 4, 6, 8, ..... 2n,

$f: \mathbb{N} \rightarrow \mathbb{R}$

$f(n) = 2n$

Notation

A sequence can be represented by

Writing the rule that provides $n^{th}$ term of the sequence.

$a_1, a_2 \ldots a_n, \ldots$

"terms: $a_i' s$"

sequence of even integers can be described by writing

$a_n=2 n$

n = 1, 2, 3......

list the terms

{$a_1, a_2, \ldots, a_n, \ldots$}

{a_n}$_{n=1}^{∞}$

Rabbit problem

1, 1, 2, 3, .......

$a_n=a_{n-1}+a_{n-2} \ n≥2$

$a_0=1, a_1=1$

$\quad \longrightarrow$ recurrence relation

sequence is different from set.

Seq {2, 4, 6, 8 .....}

≠ Seq {4, 2, 8, 6 .....}

list the terms

{${a_1, a_2, \ldots ., a_n, \ldots .}$}

{a_n}$_{n=1}^{∞}$

Sequence {a_n}$_{n=1}^{∞}$

{$={1, \frac{1}{2}, 1, \frac{1}{3}, 1, \frac{1}{4} \cdots \cdots}$}

$a_ {2 n-1}=1 \ \forall n \in \mathbb{N}, \ \ \ a_{2 n}=\frac{1}{n+1} \ \forall n \in \mathbb{N}$

$a_{2.1}=\frac{1}{2} \ \ a_2=\frac{1}{2}$

$a_{2.2}=\frac{1}{3} \ \ a_4=\frac{1}{3}$

$a_{2.3}=\frac{1}{4} \ \ a_6=\frac{1}{4}$

{${a_n}$}

[A seq. $f: \mathbb{N} \rightarrow \mathbb{R}]$

12, 14, 16, 18, .......

$a_n$ = 10 + 2n

{a_n}$_{n=1}^{∞}$ ; $a_n$ = 10 + 2n

{b_n}$_{n=6}^{∞}$ $b_n$ = 2n

$b_6$ = 2.6 = 12

$b_7$ = 2.7 = 14

$\mathbb{N} \rightarrow$ {${n_0, n_0+1....}$}

{${a_1, a_2, a_3 \ldots . }$}

$a_i'$ s - terms.

Sequences with only finite number of terms are called finite sequences.

Sequences with infinite number of terms are called infinite sequences.

[A seq. $f: \mathbb{N} \rightarrow \mathbb{R}]$

12, 14, 16, 18, .......

$a_n$ = 10 + 2n

{a_n}$_{n=1}^{∞}$ ; $a_n$ = 10 + 2n

Definition:

A real sequence is defined as a function from set of natural numbers $\mathbb{N}$ to $\mathbb{R}$ or a function from a subset {1, 2,... k} of $\mathbb{N}$ to $\mathbb{R}$.