mathematics-class-11-unit-10-chapter-01-sequence-and-series-l-1-10-vdysughse7s

### <Success style="font-size:25px"> Sequence And Series L-1 </Success>
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- A sequence of events
- A series of Mathematics tests
- Cricket test match Series.
- Succession of events

- **In Mathematics**

- "Sequence",

- "Series"

- have different technical meaning.

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<footer style = "font-size: 18px"> $\rightarrow$ Sequence & series $\rightarrow$ <span style = "color:blue"> Introduction </span> $\rightarrow$ Sequence $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Sequence And Series L-1 </Success>
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- 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?

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<footer style = "font-size: 18px">Sequence & series $\rightarrow$ Introduction $\rightarrow$ <span style = "color:blue"> Sequence </span> $\rightarrow$ Solution $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-1 </Success>
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- 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

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### <Success style="font-size:25px"> Sequence And Series L-1 </Success>
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- 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 $=(1+\frac{1}{20})+\frac{1}{20}(1+\frac{1}{20})$  
- $=(1+\frac{1}{20})^2$

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- 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"

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<footer style = "font-size: 18px">Example $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Sequence $\rightarrow$ Definition </footer>

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- 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.  
  
- $1 \rightarrow a_1$

- $2 \rightarrow a_2$

- $3 \rightarrow a_3\ldots\ldots$

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

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

- $f(n)=a_n$

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- 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: $a_i' s$"

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- 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<sub>n</sub>}$_{n=1}^{∞}$

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<footer style = "font-size: 18px">Definition $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Remark </footer>

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- 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<sub>n</sub>}$_{n=1}^{∞}$

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- Sequence {a<sub>n</sub>}$_{n=1}^{∞}$

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

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

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

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

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

- {${a_n}$}

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<footer style = "font-size: 18px">Example $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> Remark </span> $\rightarrow$ Remark $\rightarrow$ Finite and infinite sequences </footer>

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

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

- $a_n$ = 10 + 2n

- {a<sub>n</sub>}$_{n=1}^{∞}$ ; $a_n$ = 10 + 2n

- {b<sub>n</sub>}$_{n=6}^{∞}$  $b_n$ = 2n

- $b_6$ = 2.6 = 12

- $b_7$ = 2.7 = 14

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

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

- $a_i'$ s - terms.

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

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

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

- $a_n$ = 10 + 2n

- {a<sub>n</sub>}$_{n=1}^{∞}$ ; $a_n$ = 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 $\mathbb{R}$.

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### <Success style="font-size:25px"> Sequence And Series L-1 </Success>
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- Informal definition sequence
- Formal definition
- Describing sequence
- {a<sub>n</sub>}$_{n=1}^{∞}$ ; {${a_1, a_2, ....}$}
- recursive definition
- Difference between sequence and sets

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<footer style = "font-size: 18px">Finite and infinite sequences $\rightarrow$ Real sequence $\rightarrow$ <span style = "color:blue"> Summary </span> $\rightarrow$ Thank you $\rightarrow$  </footer>