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- Question 9 & 10 Exercise 4.3
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... ots, 693$$.\\ Here, $a=306$,\\ $$d=(315-306) = 9 \text { and } a_n=693 .$$\\ Let the number of terms be $n$. Then\\ \begin{align}a_n&=a_1+(n-1) d \text { becomes } \\ \Rightarrow a_1+(n-1) d&=693 \\ \R
- Question 2 & 3 Exercise 4.4
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... he sequence. ====Solution==== Here $$a_3=27 \quad\text{and}\quad a_5=243$$ and we know\\ \begin{align}a_... _1 r^2}&=\dfrac{243}{27}=9 \\ \Rightarrow r^2&=9 \text { or } r= \pm 3 .\end{align} Putting this in (i),
- Question 5 & 6 Exercise 4.3
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... \\ \Rightarrow 20 d^2&=20 \\ \Rightarrow d^2&=1 \text { or } d= \pm 1\end{align} When $a=5$ and $d=1$ t... align} a-3d&=5-3=2, \\ a-d&=5-1=4, \\ a+d&=5+1=6 \text { and } \\ a+3 d&=5+3=8.\end{align} When $a=5$ an
- Question 5 & 6 Exercise 4.5
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... then\\ $$S_{10}=\dfrac{a_1(r^{10}-1)}{r-1} \quad \text{and}\quad S_5=\dfrac{a_1(r^5-1)}{r-1}$$\\ Putting... \\ \Rightarrow(r^5-1)(r^5-243)&=0 \\ \Rightarrow \text { Either } r^5-10&=0 \text { or } r^5-243=0 \\ \R
- Question 4 Exercise 4.5
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... verline{8}&=0.8+0.08+0.008 \div 0.0008+ \ldots\\ \text { or } 0 . \overline{8}&=0.8+(0.1)(0.8) +(0.1)^2(... \overline{63}&=1+0.63+0.0063+0.000063 +\ldots \\ \text { or } 1 . \overline{63}&=1+[0.63+ (0.01)(0.63)-(
- Question 1 Exercise 4.5
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... =a_1 r^{n-1}$$,\\ \begin{align}3.2^9&=3(2)^{n-1} \text { or }(2)^{n-1}=\dfrac{3.2^9}{3} \\ \Rightarrow(2)^{n-1}&=2^9 \\ \Rightarrow n-1&=9 \text { or } n=10 \\ \text {. Now }\quad S_n&=\dfrac{a
- Question 7 & 8 Exercise 4.5
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... ightarrow a^3&=1728\\ \Rightarrow \quad a&=12,\\ \text{putting}\text{in} (1)\\ \dfrac{12}{r}+12+12 r&=38\\ \Rightarrow \dfrac{1}{r}+1+r&=\dfrac{38}{12}=\dfrac
- Question 8 Exercise 4.2
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... {b+c-a}{a}&=\dfrac{a+b-c}{c}-\dfrac{c+a-b}{b} \\ \text{Let}\quad S&=\dfrac{a+b+c}{2} \\ \Rightarrow a+b+c&=2 S\\ \text{then} \Rightarrow a+b-c&=2(S-c) \text {, }\\ a+c-
- Question 3 & 4 Exercise 4.3
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... Rightarrow 5 n&=350-20=330 \\ \Rightarrow n&=66, \text { now for the sum } \\ S_n&=\dfrac{n}{2}(a_1+a_n), \text { that becomes } \\ S_{66}&=\dfrac{66}{2}(25+350)
- Question 11 & 12 Exercise 4.3
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... d $a_1=16 \mathrm{ft}$\\ The distance in the $2^{\text {nd }}$ second $a_2=48 \mathrm{ft}$\\ The distanc... the thirtieth day? ====Solution==== \begin{align}\text{Saves at first day}& =R s .1\\ \text{Saves at sec
- Question 6 Exercise 4.1
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... uences is $$P_0=1, P_{r+1}=\dfrac{n-r}{r+1} P_r, \text{ where } r=0,1,2,3,\ldots.$$ =====Question 6(i)=... follows: $$P_0=1, P_{r+1}=\dfrac{5-r}{r+1} P_r, \text{ where } r=0,1,2,3,\ldots.$$ For $r=0$ \begin{al
- Question 13 & 14 Exercise 4.3
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... re in the theater? ====Solution==== \begin{align}\text{Total number of rows}& n=40,\\ \text{Seats in a first row} a_1&=20\\ \text{Seat in a second row} a_2&=2
- Question 3 Exercise 4.5
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... 1 r}&=\dfrac{1}{2}\\ \Rightarrow r&=\dfrac{1}{2} \text {, }\end{align} putting this in (i), we have\\ \b... gin{align}\dfrac{a_1}{2}&=2\\ \Rightarrow a_1&=4 \text {. }\\ a_2&=a_1 r=4 \cdot \dfrac{1}{2}=2,\\ a_3&
- Question 9 & 10 Exercise 4.5
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... three terms is:\\ $$S_3=\dfrac{a_1(r^3-1)}{r-1} \text {. }$$\\ The sum of the $3$ terms is $9$ times th... ghtarrow(r^3-1)(r^3-8)&=0 \\ \Rightarrow r^3&=1 \text { or } r^3=8 \\ \Rightarrow r&=1 \text { or } r=
- Question 11 & 12 Exercise 4.5
- f Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. ==... ^{p-1})^{q-r} . \\ b^{r-p}&=(a_1 r^{q-1})^{r-p}, \text { and } \\ c^{p-q}&=(a_1 r^{r-1})^{p-q}.\end{alig... r+q r-p q r+p+p r-q r-p+q} \\ & =1 \cdot r^0=1 . \text { Thus } \\ a^{q-r} b^{r-p} c^{p-q}&=1.\end{align