mathematics-class-11-unit-10-chapter-05-sequence-and-series-l-5-10-ymhsqjmj3bs

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Arithmetic progression (AP) </primary>
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- A sequence  $(a_n)^{\infty}_{n=1}$  is called an AP, if

- $ a_{n+1}=a_n+d \ \ \forall n \geqslant 1$  
  
- i.e. If $\\{a_ n\\}_ {n=1}^{\infty}$ is an AP with $a_ 1 =a$, called first term and common difference d, then AP can be written in standard form as  
- $a, a+d, a+2 d, \cdots$

- $n^{\text {th }} \text { term of A.P } =a+(n-1) d$

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<footer style = "font-size: 18px"> $\rightarrow$ Sequence & series lecture-5 $\rightarrow$ <span style = "color:blue"> Arithmetic progression (AP) </span> $\rightarrow$ Property $\rightarrow$ Property </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Property </primary>
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- **1)**  If $\left(a_n\right) _{n=1}^{\infty}$  is an AP, then sequence

- $\left(b_n\right)_{n=1}^{\infty}$ obtained by $b_n=a_n+k \ \ \forall n$ is an AP.

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<footer style = "font-size: 18px">Sequence & series lecture-5 $\rightarrow$ Arithmetic progression (AP) $\rightarrow$ <span style = "color:blue"> Property </span> $\rightarrow$ Property $\rightarrow$ Property </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Property </primary>
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- **2)** If ${(a_ n)}_ {n=1}^{\infty}$ is an AP, then the Sequence $(b_ n)_{n=1}^{\infty}$ obtained by subtracting a constant "$k$" to each term is again an AP.

- $b_n=a_n-k \ \ \forall n \geqslant 1$

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<footer style = "font-size: 18px">Arithmetic progression (AP) $\rightarrow$ Property $\rightarrow$ <span style = "color:blue"> Property </span> $\rightarrow$ Property $\rightarrow$ Property </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Property </primary>
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- **3)** If $\left(a_n\right)_{n=1}^{\infty}$ is an AP, then sequence obtained by multiplying each term of $\\{a_n\\}$ with a constant is again an AP.

- $\left(a_n\right)_{n=1}^{\infty} \text { is an } A P$

- $a_{n+1}-a_n=d$

- $\text { Consider }\left(b_n\right)_{n=1}^{\infty} \ \ \forall n \in \mathbb{N}$    
- $b_{n+1}-b_n$  
- $ \ \=c a_{n+1}-c a_n$  
- $\ \ =c\left(a_{n+1}-a_n\right)$  
- $ \quad =c d \ \ \forall n \in \mathbb{N}$

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<footer style = "font-size: 18px">Property $\rightarrow$ Property $\rightarrow$ <span style = "color:blue"> Property </span> $\rightarrow$ Property $\rightarrow$ Question 1 </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Property </primary>
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- **4)** If $\left(a_ n\right)_{n=1}^{\infty}$ is an AP, then sequence
- ($b_n$) obtained by dividing each term of $\left(a_n\right)$ with $a$
- non-zero constant remains to be an AP.

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<footer style = "font-size: 18px">Property $\rightarrow$ Property $\rightarrow$ <span style = "color:blue"> Property </span> $\rightarrow$ Question 1 $\rightarrow$ Arithmetic mean </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Question 1 </primary>
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- Let $a, b$ be given numbers. Can we "insert" a number $A$ such that $a, A, b$ are terms of an AP ?

- **Solution:**

- $
\begin{aligned}
& A-a=b-A \\\ 
\Rightarrow & 2 A=a+b \\\ 
\Rightarrow & A=\frac{a+b}{2}
\end{aligned}
 $

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<footer style = "font-size: 18px">Property $\rightarrow$ Property $\rightarrow$ <span style = "color:blue"> Question 1 </span> $\rightarrow$ Arithmetic mean $\rightarrow$ Question 2 </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Arithmetic mean </primary>
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- Given two numbers $a, b$ the number $\frac{a+b}{2}$ is called
- Arithmetic Mean (AM) of $a$ and $b$.

- $A M(a, b)=\frac{a+b}{2}$

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<footer style = "font-size: 18px">Property $\rightarrow$ Question 1 $\rightarrow$ <span style = "color:blue"> Arithmetic mean </span> $\rightarrow$ Question 2 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Question 2 </primary>
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- Let $a$ and $b$ be two numbers given.

- Can we insert numbers
- $A_1, A_2\cdot\cdot\cdot A_n$,
- So that
- $a, A_1, A_2$.. $A_n, b$ are successive terms of a sequence which is AP?

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<footer style = "font-size: 18px">Question 1 $\rightarrow$ Arithmetic mean $\rightarrow$ <span style = "color:blue"> Question 2 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $a, A_1, A_2 \ldots A_n, b$ are in AP.

- The $(n+2)$ th term $=a+(n+2-1) d$

- $b=a+(n+1) d$  
- $\Rightarrow d=\frac{b-a}{n+1}$  
- Then
- $A_1=a+d=a+\frac{b-a}{n+1}$  
- $A_2=a+2 d=a+2\frac{(b-a)}{n+1}$

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<footer style = "font-size: 18px">Arithmetic mean $\rightarrow$ Question 2 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Remarks </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Remarks </primary>
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- AP, $a_{n+1}-a_n=d \$ $\forall n \in \mathbb{N}$
- Std form $a, a+d, a+2 d +\cdot\cdot\cdot\cdot\cdot$
- $n^{\text {th }}$ term $a_n=a+(n-1) d$
- Given two numbers $a$ and $b$,

- $\operatorname{AM}(a, b)=\frac{a+b}{2}$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Remarks </span> $\rightarrow$ Question 3 $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $\begin{array}{ccccc}
 1 & 2 & 3 & 4 & 5 \\\ 
\cdots & & 100 \\\ 
 1 & 2 & 3 & 4 & 5
\end{array}$

- 100, 99, 98........

- Total sum $=50 \times 101$

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<footer style = "font-size: 18px">Remarks $\rightarrow$ Question 3 $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Question 3 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Question 3 </primary>
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- What is sum of $n$ terms of the AP $a, a+d, a+2 d+\cdot\cdot ?$  
 
- **Solution:**

- $S_n =a+(a+d) +\ldots+a+(n-1) d$  
- $n^{\text {th }} \text { term } a+(n-1) d =l$  
 
- $S_n=a+a+d+\ldots+l$

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<footer style = "font-size: 18px">Question 3 $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Question 3 </span> $\rightarrow$ Solution $\rightarrow$ Geometric progression </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Geometric progression </primary>
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- A sequence $\left(a_n\right)_{n=1}^{\infty}$
- is Said to be a Geometric progression $(G P)$ if no terms are zero and

- $\begin{aligned} & \frac{a_{n+1}}{a_n}=r \\ & \forall n \in \mathbb{N}\end{aligned}$  
- $r$ is called common ratio.

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<footer style = "font-size: 18px">Question 3 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Geometric progression </span> $\rightarrow$ Example $\rightarrow$ Notes </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $
\begin{aligned}
& 3,6,12,24, \ldots . \\\ 
& \frac{a_{n+1}}{a_n}=2 \quad \forall n \in \mathbb{N} \\\ 
& \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \cdots \frac{1}{2 n} ;
\end{aligned}
 $

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<footer style = "font-size: 18px">Solution $\rightarrow$ Geometric progression $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Notes $\rightarrow$ Sum of n terms </footer>

### <Success style="font-size:25px"> Sequence And Series L-5 </Success>
### <primary style="font-size:24px"> Notes </primary>
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- If first term is " $a$ " and common ratio is " r " then we can write GP (in std form) as

- $a, r a, r^2 a, \ldots . .$

- $\begin{gathered}n^{\text {th }} \text { term } \\ =a r^{n-1}\end{gathered}$

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<footer style = "font-size: 18px">Geometric progression $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Notes </span> $\rightarrow$ Sum of n terms $\rightarrow$ Thank you </footer>