### <Success>Sequence & series lecture 4</Success> <div style='float: left; width:50%;'> </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-09-sequence-and-series-l-9-10-ki2eIuhnusq-2.jpg"></p>

### <Success>Problem 1</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>You drop a ball</p> <p>From “a” meters above a flat surface. Each time the ball hits the surface after falling a distance h, it rebounds a distance rh, where r is positive but less than 1.</p> <p>Find the total distance the ball travels up and down. Find the total no. of seconds the ball is travelling assuming a = 4 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-09-sequence-and-series-l-9-10-ki2eIuhnusq-3.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-3a.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-3b.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:50%;font-size:30px;text-align:left'> <p>Total distance S</p> <p>$= a + ra + ra + r^2a + r^2a + ……….$<br> $= a + 2ra + 2r^2a + ……….$</p> <p>GP with first term 2ra and common ratio r.</p> <p>$=a+\frac{2 r a}{1-r}$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/88dece00-369a-4c64-22e3-ff347c60cb00/public"></p> <!----> <!---->

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$ S=u t+\frac{1}{2} g t^2$</p> <p>$S=\frac{1}{2} g t^2(\because u=0)$</p> <p>$S=4.9 t^2$</p> <p>$\Rightarrow t=\sqrt{S / 4.9}$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-6.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-6a.jpg">

### <Success>Problem 2</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Can you make an infinite series of non-zero terms that converge to any number you want?</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-7.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Let L be a given number.</p> <p>Let $a, ar, ar^2, ……$</p> <p>be a geometric progression.</p> <p>$a+ar +a r^2+\ldots .$</p> <p>is convergent for |r| < 1</p> <p>$\sum_{n=1}^{\infty} a r^{n-1}=\frac{a}{1-r}$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-8.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-8a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$\frac{a}{1-r}=L$,</p> <p>where</p> <p>$|r| < 1$</p> <p>Thus selecting r such that $|r| < 1$, and $a = L(1 - r)$</p> <p>$L(1 - r) + Lr(1 - r) + Lr^2(1 - r) + ……..$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-9.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-9a.jpg"></p>

### <Success>Problem 3</Success> <div style='float: left; width:50%;font-size:30px;text-align:left'> <p>Find the sum of areas of all squares.</p> <p>Outer most square has area<br> $ A = 4m^2$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/b85b94c4-33ab-42ad-3c19-d1cc5daeb400/public"></p> <!---->

### <Success>Solution</Success> <div style='float: left; width:50%;font-size:30px;text-align:left'> <p>BC = $\sqrt{A B^2+A C^2}$</p> <p>$ =\sqrt{(\frac{a}{2})^2+(\frac{a}{2})^2}$</p> <p>$=\sqrt{\frac{2 a^2}{4}}$</p> <p>$=\frac{a}{\sqrt{2}}$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/d73b4d46-9c6d-46c5-319c-bc758696d400/public"></p> <!----> <!--</div>--> <!-- <div style='float: right; width:0%;'>--> <!-- -->

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Areas would be</p> <p>$a^2,(\frac{a}{\sqrt{2}})^2,(\frac{\frac{a}{\sqrt{2}}}{\sqrt{2}})^2, \ldots$</p> <p>ie $a^2, \frac{a^2}{2}, \frac{a^2}{4}, \ldots$.</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-12.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Sum of Areas</p> <p>$=a^2+\frac{a^2}{2}+\frac{a^2}{4}+\cdots$</p> <p>$=\frac{a^2}{1-\frac{1}{2}}=2 a^2$</p> <p>$a^2=4$</p> <p>Sum of areas</p> <p>$= 2 \times 4$</p> <p>$= 8 m^2$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-13.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-13a.jpg"></p>

### <Success>Problem 4</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Second term of a GP is 1000 and common ratio is $\frac{1}{n}$, where $n \in \mathbb{N}$</p> <p>Let “$P_n$” be product of “n” terms. If $P_6 > P_5$ and $P_6>P_7$, what is sum of all possible values of n.</p> <p><strong>Note that</strong> all terms of this GP are positive.</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-14.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-14a.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-14b.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$P_6$ = Product of six terms.</p> <p>$=P_5 \cdot t_6$</p> <p>$t_n =n^{th}$ term</p> <p>Given $P_6 > P_5$</p> <p>$t_6>1$ <br> $P_7=P_6 \cdot t_7$</p> <p>Given $P_7<P_6 \quad \therefore t_7<1$</p> <p>Thus $t_6>1$ and $t_7<1$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-15.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-15a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:26px;text-align:left'> <p>A G.P $a, ar, ar^2, ar^3$</p> <p>$ a\rarr t_1$,</p> <p>$ ar\rarr t_2$,</p> <p>$ ar^2\rarr t_3$,</p> <p>$ ar^3\rarr t_4$, …….</p> <p>$\therefore t_6$ = Six term</p> <p>= second term. $r^4$</p> <p>= 1000 $r^4$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-16.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-09-sequence-and-series-l-9-10-ki2eIuhnusq-16a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$t_7$ = Seventh terms</p> <p>$=$ Second term. $r^5$</p> <p>$= 1000 r^5$</p> <p>$t_6>1 \Rightarrow 1000 r^4>1$</p> <p>$\Rightarrow r^4>\frac{1}{1000}$</p> <p>$\Rightarrow \frac{1}{n^4} >\frac{1}{1000}$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-17.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$\Rightarrow n^4<1000$</p> <p>$t_7<1 \Rightarrow 1000 r^5<1$</p> <p>$\Rightarrow 1000 \frac{1}{n^5}<1$</p> <p>$\Rightarrow n^5>1000$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-18.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Thus we search for all possible values of n such that $n^4<1000$ and $n^5>1000$.</p> <p>ie, $n < 6$ and $n \geqslant 4$<br> Thus possible values of $n$ are $4$ and $5$.</p> <p>required sum $= 9.$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-19.jpg"></p>

### <Success>Problem 5</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Sum of first 12 terms of a G.P is equal to sum of first 14 terms of same GP .<br> Given that sum of first 17 terms is 92. What is the third term in GP?</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-21.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$S_{12} = S_{14}$</p> <p>$S_{14} = S_{12}+t_{13}+t_{14}$</p> <p>$\Rightarrow S_{12} = S_{12}+t_{13}+t_{14}$</p> <p>i.e.</p> <p>$t_{13}+t_{14}=0.$</p> <p>i.e. $t_{13}+r t_{13}=0.$</p> <p>i.e. $t_{13}(1+r)=0.$</p> <p>i.e. $t_{13}=0$ or $r = -1$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-23.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$t_{13} = 0$ is NOT possible</p> <p>$\therefore r = -1$</p> <p>Let the first term be a</p> <p>$a, ar, ar^2, ……$</p> <p>$= a, -a, a, -a, …….$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-25.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$\therefore$ Sum of n terms of this G.P</p> <p>$= 0$ if n is even</p> <p>$= a$ if n is odd</p> <p>Sum of first $17$ terms $= 92$</p> <p>$a = 92$</p> <p>$\therefore$ Third term $= 92$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-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-09-sequence-and-series-l-9-10-ki2eIuhnusq-26a.jpg"></p>

### <Success>Problem 6</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>The sum of first $25$ terms of an AP is $525$ and the sum of next $25$ terms is $725.$ What is the common difference?</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-27.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>Let $t_1, t_2 \cdots t_{25}$,</p> <p>$t_{26}, \ldots t_{50}, \ldots$ be the terms of given A.P.</p> <p>Given that</p> <p>$S_{25} =\sum_{i=1}^{25} t_i=525$</p> <p>$K_{25} =t_{26}+t_{27}+\cdots+t_{50}$</p> <p>$= 725$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-28.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$t_{26} =t_1+25 d$</p> <p>$t_{27} =t_1+26 d$</p> <p>$=t_1+d+25 d$</p> <p>$=t_2+25 d$</p> <p>$t_{28} =t_1+27 d$</p> <p>$=t_1+2 d+25 d$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-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-09-sequence-and-series-l-9-10-ki2eIuhnusq-29a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$t_{28}=t_3+25 d$<br> $\vdots$</p> <p>$t_{50}=t_{25}+25 d$</p> <p>$k_{25} = t_{26}+\ldots+t_{50}$</p> <p>$=(t_1+25 d)+(t_2+25 d)$</p> <p>$+\cdots+(t_{25}+25 d)$</p> <p>$=(t_1+t_2+\cdots+t_{25})+ \underbrace {25d + 25d + ….. + 25d} _{25 \text{terms}}$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-30.jpg"></p> <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-09-sequence-and-series-l-9-10-ki2eIuhnusq-30a.jpg">

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;text-align:left'> <p>$k_{25} = S_{25}+25.25 d$</p> <p>$\Rightarrow d =\frac{k_{25}-S_{25}}{625}$</p> <p>$=\frac{725-525}{625} \\ = \frac{200}{625} \\ = \frac{8}{25}$</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-09-sequence-and-series-l-9-10-ki2eIuhnusq-31.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>