### <Success>Sequence & series lecture-7</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-2.jpg">

### <Success>Recall</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <ul> <li>Given two numbers “$a$” and “$b$”, $A M$ of $a$ and $b$ is defined by $AM (a, b) = \frac{A + B } {2}$</li> </ul> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-3.jpg"></p>

### <Success>Recall</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <ul> <li>Given two positive numbers " $a$ " and " $b$ " the GM of $a, b$ is defined as</li> </ul> <p>$\operatorname{GM}(a, b)=\sqrt{a b} \text {. }$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-4.jpg">

### <Success>Example</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$ \operatorname{AM}(1,2)=\frac{1+2}{2}=1.5 $</p> <p>$\operatorname{GM}(1,2)=\sqrt{2} $</p> <p>$ \operatorname{AM}(1,4)=2.5 $</p> <p>$ \operatorname{GM}(1,4)=\sqrt{4}=2 $</p> <p>$ \operatorname{AM}(2,8)=5 $</p> <p>$ \operatorname{GM}(2,8)=4$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-5.jpg"></p>

### <Success>Question 1</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Let $a, b$ be two positive reals.</p> <p>$ \operatorname{AM}(a, b)=\frac{a+b}{2} $</p> <p>$ \operatorname{GM}(a, b)=\sqrt{a b.}$<br> $\quad A M \geqslant G M \ \ \rightarrow ?$.</p> <p>$\quad \frac{a+b}{2} \geqslant \sqrt{a b} \ \ \rightarrow ?$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-6.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>consider</p> <p>$\frac{a+b}{2}-\sqrt{a b}$<br> $ =\frac{a+b-2 \sqrt{a b}}{2} $</p> <p>$ =\frac{(\sqrt{a}-\sqrt{b})^2}{2} $</p> <p>$ \geqslant 0$</p> <p>$ A M \geqslant G M $</p> <p>$\frac{a+b}{2} \geqslant \sqrt{a b}$</p> <p>Equality holds iff $a=b$.</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-7.jpg"></p>

##### <Success>Arithmetic and geometric mean relation</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>consider a rectangle with Side lengths " $a$ " and " $b$ "</p> <p>Perimeter $=2 a+2 b$</p> <p>Area $=a b$</p> <p>Consider square of Side lengths</p> <p>$\sqrt{a b}$.</p> <p>$\text { Area } =a b $</p> <p>$\text { Perimeter } =4 \sqrt{a b} $</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-9.jpg">

##### <Success>Arithmetic and geometric mean relation</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p><strong>AM - GM inequality gives</strong></p> <p>$\frac{a+b}{2} \geqslant \sqrt{a b} .$</p> <p>$2 a+2 b \geqslant 4 \sqrt{a b}$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-11.jpg">

##### <Success>Arithmetic and geometric mean relation</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Given</p> <ul> <li> <p>“$n$” real numbers $a_1, a_2 \ldots a_n$. The $AM$ of these numbers is defined as $A M\left(a_1, a_2, a_n\right)=\frac{a_1+a_2+ ~+a_n}{n}$</p> </li> <li> <p>“$n$” positive reals $a_1, a_2, . . a_n$. the Geometric mean of these numbers. $\operatorname{GM}\left(a_1, a_2 \quad a_n\right) = \left(a_1 a_2 \ldots a_n\right)^{\frac{1}{n}}$</p> </li> </ul> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-12.jpg">

##### <Success>Arithmetic and geometric mean relation</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>i.e.</p> <p>Given “$n$” positive reals $a_1, a_2, \ldots a_n$</p> <p>$ A M \geqslant G M $</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-14.jpg"></p>

### <Success>Infinite arithmetic series</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Given an $ A P $</p> <p>$a, a+d, a+2 d, . a+(n-1) d,….$</p> <p>$\sum_{n=1}^{\infty}[a+(n-1) d]$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-15.jpg"></p>

### <Success>Observation</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$Let \sum_{n=1}^{\infty} a_n \text{~be convergent.}$</p> <p>i.e., sequence of partial sum $S_n=a_1+a_2+\ldots+a_n$ is convergent.</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-16.jpg">

### <Success>Convergence of infinite arithmetic series</Success> <div style='float: left; width:100%;font-size:28px;text-align:left'> <p>$\underset{n \rightarrow \infty}{L t} \quad S_n= S$</p> <p>As " $n$ " becomes large both $S_n$ and $S_{n-1}$ are close to $S.$</p> <p>$ a_n=S_n-S_{n-1} $</p> <p>$ \underset{n \rightarrow \infty}{L t} a_n=0$</p> <p>$\sum_{n=1}^{\infty} a_n \text{~is convergent}$</p> <p>$\Rightarrow \quad a_n \rightarrow 0$<br> as $n \rightarrow \infty$.</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-17.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-17a.jpg"></p>

### <Success>Convergence of infinite arithmetic series</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$P \Longrightarrow \ Q$ is logically</p> <p>equivalent to</p> <p>Not $Q \Longrightarrow$ Not $P$.</p> <p>If $a_n \not\longrightarrow 0$ as $n \rightarrow \infty$ then $\sum a_n$ is NOT convergent.</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-18.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-18a.jpg"></p>

### <Success>Convergence of series</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Given an AP</p> <p>$a, a+d, a+2 d, \cdots a+(n-1) d$;</p> <p>$\sum_{n=1}^{\infty}[a+(n-1) d]$</p> <p>$ n^{\text {th }} \text { term } =a+(n-1) d $</p> <p>$ d \neq 0 $</p> <p>$\underset{n \rightarrow \infty}{L t} a+(n-1) d = \pm \infty $</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-19.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-19a.jpg"></p>

### <Success>Convergence of series</Success> <div style='float: left; width:100%;font-size:28px;text-align:left'> <p>$\therefore\sum_{n=1}^{\infty} a+(n-1) d$ is NOT convergent.</p> <p>$\underline{d = 0}$</p> <p>$ {n^{\text {th }} \text { term }}=a $</p> <p>$ \underset{n \rightarrow \infty}{L t} \quad a \neq 0 $</p> <p>$ \quad(\text { if } a \neq 0)$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-20b.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-20.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-20a.jpg">

### <Success>Convergence of series</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$\sum_{n=1}^{\infty} a_n \text { is convergent } $</p> <p>$ \rArr a_n \text{ is close to zero as }$</p> <p>“$n$” $\text{becomes large}$</p> <p>$ \underset{n \rightarrow \infty}{L t} a_n = 0 .$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-21.jpg"></p>

### <Success>Remark</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$\underline{\text{Remark}}$</p> <p>If $a_n \longrightarrow 0$ as $n \longrightarrow \infty$, then we cannot conclude that $\sum a_n$ is convergent.</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-22.jpg"></p>

### <Success>Example</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$ 1, \frac{1}{2}, \frac{1}{3}, \cdots \frac{1}{n}$,…..</p> <p>$ \sum \frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}+$……</p> <p>$ S_{2^n} \geqslant 1+\frac{n}{2} $</p> <p>$\sum \frac{1}{n}$ is NOT Convergent</p> <p>But $\frac{1}{n} \longrightarrow 0$</p> <p>as $n \rightarrow \infty$.</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-23.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-23a.jpg"></p>

### <Success>Example</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$</p> <p>$\sum_{n=1}^{\infty} \frac{1}{2^{n-1}}\text{ is convergent}$</p> <p>$\frac{1}{2^{n-1}} \longrightarrow 0$</p> <p>$\text{as} ~n \rightarrow \infty$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-24.jpg"></p>

### <Success>Question 2</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Sum of $n$ terms of two $AP$ are in the ratio $\frac{3n+8}{7n + 15}.$</p> <p>Find the ratio of their $12^{\text{th}}$ terms.</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-25.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-25a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Let us assume that first $AP$ has first term $a_1$ and common difference $d_1$</p> <p>$a_1, a_1 + d_1, a_1 + 2d_1,……$</p> <p>Let second $AP$ has first term $a_2$ and common difference $d_2$</p> <p>$a_2, a_2 + d_2, a_2 + 2d_2,…..$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-26.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-26a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$n^{\text{th}}$ term $I$ $AP$</p> <ul> <li>$ = a_1 + (n-1) d_1$</li> </ul> <p>$n^{\text{th}}$ term $II$ $AP$</p> <ul> <li>$ a_2 + (n-1) d_2 $</li> </ul> <p>$ \text{sum of}$ “$n$” terms of $I$ $AP$</p> <ul> <li>$= \frac{n}{2} [2. a_1 + (n-1) d_1]$</li> </ul> <p>$ \text{sum of}$ “$n$” terms of $II$ $AP$</p> <ul> <li>$= \frac{n}{2} [2. a_2 + (n-1) d_2]$</li> </ul> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-27.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$\frac{\frac{n}{2}(2a_1 + (n-1) ~d_1)}{\frac{n}{2}(2a_2 + (n-1) ~d_2)}=\frac{3_n+8}{7_n+15}$</p> <p>$\frac{2a_1 + (n-1) ~d_1}{2a_2 + (n-1) ~d_2}=\frac{3_n+8}{7_n+15}$</p> <p>$12^{\text{th}}$ term of $I$ $AP$</p> <ul> <li>$= a_1 + 11 ~d_1$</li> </ul> <p>$12^{\text{th}}$ term of $II$ $AP$</p> <ul> <li>$= a_2 + 11 ~d_2$</li> </ul> <p>$\frac{a_1 + 11 ~d_1}{a_2 + 11 ~d_2} = ?$</p> <p>$\frac{2a_1 + 22 ~d_1}{2a_2 + 22 ~d_2} = ? …(*)$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-29.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-29a.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-29c.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-29b.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$\text{Putting} ~n= 23 \ \text{in} ~(*)$</p> <p>$\frac{2a_1 + 22 ~d_1}{2a_2 + 22 ~d_2} = \frac{3.23 + 8}{7.23+15}$</p> <p>$ = \frac{7}{16}$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-10-chapter-07-sequence-and-series-l-7_10-uzxd9jrazpy-30.jpg"></p> </div> <p>======</p> <h3 id="successthank-yousuccess"><Success>Thank you</Success></h3> <div style='float: left; width:100%;font-size:30px;text-align:left'> </div>