mathematics-class-11-unit-10-chapter-02-arithmetic-geometric-harmonic-progre-l-2-2-xeeperigzs

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $ a, b, c \rightarrow A P $

- $ a=p-r $

- $ b=p \text { for some } p, p $

- $ c=p+r $  
  
- $a+b+c= \frac{3}{2}$  
- $\Rightarrow p-r+p+p+r= \frac{3}{2}$  
- $\Rightarrow p= \frac{1}{2}$

- $a=p-r =\frac{1}{2}-r$ 
- $ a^2, b^2, c^2 \rightarrow \text{G P}$  
- $(p-r)^2, p^2,(p+r)^2 \rightarrow \text{G P}$  
- $(p-r)^2(p+r)^2=p^4$

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<footer style = "font-size: 18px">Arithmetic geometric & harmonic progressions lecture -2 $\rightarrow$ Question 8 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- The sum of first $(n-1)$ terms

- $ \quad=c(n-1)^2 $

- $ n-t h \text { term }=c n^2-c(n-1)^2=c(2 n-1)$  
  
- call $r-$ th term $a_r$

- $\sum_{r=1}^n a_r^2  =c^2 \sum_{r=1}^n(2 r-1)^2 $

- $ =4 c^2 \sum_{r=1}^n r^2-4 c^2 \sum_{r=1}^n r+c^2 \sum_{r=1}^n 1 $

- $ =\frac{4 c^2 n(n+1)(2 n+1)}{6}-4 c^2 \frac{n(n+1)}{2}+c^2 n $

- $ =\frac{c^2 n}{3}{({(2 n+2)(2 n+1)-6 n-6+ 3 })}$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Question 9 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Question 10 </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $ a_n < 0$

- $ a_n=\frac{1}{b+(n-1) d} $

- $b+(n-1) d<0 $

- $\frac{1}{5}+(n-1)\left(-\frac{4}{25 \times 19}\right)<0 $

- $\Rightarrow 1+(n-1)\left(-\frac{4}{5 \times 19}\right)<0 $

- $\Rightarrow(n-1)>\frac{5 \times 19}{4} \Rightarrow n>{\frac{5 \times 19}{4}+1}=\frac{99}{4} $  
- $n \geqslant 25 $

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<footer style = "font-size: 18px">Question 11 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Question 12 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Question 12 </primary>
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- Suppose four distinct numbers $a_1, a_2, a_3, a_4$ are in geometric progression. Let $b_1=a_1$ and $b_i=b_{i-1}+a_i$ for $i=2,3,4$.

- **Statement 1:** The numbers $b_1, b_2, b_3, b_4$ are neither in arithmetic progression, nor in geometric progression.

- **Statement 2:** The numbers $b_1, b_2, b_3, b_4$ are in harmonic progression.

- (1) Statement 1 is false, Statement 2 is true.  
- (2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.  
- (3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.  
- (4) Statement 1 is true, Statement 2 is false.

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Question 12 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $ b_ 1=a_1$

- $ b_ 2=b_ 1+a_ 2=a_ 1+a_2$

- $ b_ 3=b_ 2+a_ 3=a_ 1+a_ 2+a_3$

- $ b_ 4=b_ 3+a_ 4=a_ 1+a_ 2+a_ 3+a_4$

- $ b_ 2-b_ 1=a_ 2 $

- $ b_ 3-b_ 2=a_ 3, \ \  a_ 2 \neq a_3 $  
 
- $ b_ 2-b_ 1 \neq b_ 3-b_ 2 \ \Rightarrow 2 b_ 2 \neq b_ 1+b_3$

- $b_ 1, b_ 2, b_3 \text { are not in AP }$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Question 12 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $a_ i=a_ 1 r^{i-1} \ \ \ \text{for} \ i=2,3,4$    
 $\quad \quad \downarrow$  
- ${\text{common ratio of} \ a_ 1, a_ 2, a_ 3, a_4}$  
- **Note**  
- $a_1 \neq 0, \ \ r \neq 0$   
  
- $b_2  =a_1+a_1 r=a_1(1+r)$

- $b_3  =a_1+a_1 r+a_1 r^2=a_1\left(1+r+r^2\right)$

- $b_4  =a_1+a_1 r+a_1 r^2+a_1 r^3 =a_1\left(1+r+r^2+r^3\right)$

- $b_1 b_3  =b_2^2\left(\text { if } b_1, b_2, b_3\right. \text { are in GP) }$

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<footer style = "font-size: 18px">Question 12 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- If $b_1, b_2, b_3, b_4$ are in HP, then

- $ \quad \frac{1}{b_1}+\frac{1}{b_3}=\frac{2}{b_2} $

- $ \text { i.e. } \frac{1}{a_1}+\frac{1}{a_1\left(1+r+r^2\right)}=\frac{2}{a_1(1+r)} $

- $ \text { i.e. } \quad \frac{1+r+r^2+1}{1+r+r^2}=\frac{2}{1+r} $

- $ \text { i.e. } \quad(1+r)\left(2+r+r^2\right)=2\left(1+r+r^2\right) $

- $ \text { i.e. } \quad 2+r+r^2+2 r+r^2+r^3 =2+2 r+2 r^2$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Question 13 </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Question 13 </primary>
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- Let $a_1, a_2, \ldots, a_{100}$ be in arithmetic progression with $a_1=3$ and $s_p=\sum_{i=1}^p a_i$ for $1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$, let $m=5 n$. If $s_m / s_n$ does not depend on $n$, then what is the value of $a_2$ ?  
  
    
- **Solution:**

- $d \rightarrow$ common difference

- $ a_ 2=a_1+d $

- $ S_ p = \sum_ {i=1}^p a_ i$

- $=p a_1+(1+2+\cdots+(p-1)) d $

- $=3 p+\frac{p(p-1)}{2} d $
- $=\frac{p}{2} (6+(p-1) d)$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Question 13 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Question 14 </primary>
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- Let $a, b, c$ be positive integers such that $b / a$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2$, then what is the value of $\frac{a^2+a-14}{a+1}$ ?

- $ a=\frac{b}{r} \quad \frac{b}{a} \in \mathbb{Z}$

- $c=b r \quad  r=\frac{b}{a}$

- $ \therefore r \in \mathbb{Z}_{>0}$

- $ \frac{a+b+c}{3}=b+2 \Rightarrow a-2 b+c=6$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Question 14 </span> $\rightarrow$ Solution $\rightarrow$ Question 15 </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Question 15 </primary>
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- Suppose that all the terms of an arithmetic progression are positive integers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6: 11$ and the seventh term of the progression lies between 130 and 140, then what is the common difference of this arithmetic progression?  
  
    
- **Solution:**

- $d \rightarrow$ common difference

- $a_r \rightarrow$ r-th term
- $ a_r=a_1+(r-1) d $

- $\forall r \geqslant 1$

- $a_r \in \mathbb{Z}_{>0} \quad \forall r \geqslant 1$

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<footer style = "font-size: 18px">Question 14 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Question 15 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Arithmetic Geometric Harmonic Progre L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $d \in \mathbb{Z}_{>0} $

- <body>
   <div id="reftest-element">$
         d = \underbrace{\left(a_1+d\right)}_{\in \mathbb{Z}_{>0}}-\underbrace{a_1}_{\in \mathcal{Z}_{>0}}
         $</div>
   </body>

- $\frac{\sum_{i=1}^7 a_i}{\sum_{i=1}^{11} a_i}=\frac{6}{11}$

- $\sum_{i=1}^7 a_i=7 a_1+{ \underbrace{(1+2+\cdots+6)} } d $

$  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \frac{6 \cdot 7}{2} $

- $=7(a_1+3 d)$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Question 15 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>