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- Question 6 Exercise 4.1
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... r, \text{ where } r=0,1,2,3,\ldots.$$ For $r=0$ \begin{align}&P_{0+1}=\dfrac{5-0}{0+1} P_0\\ \implies &P_1=5.\end{align} For $r=1$ \begin{align}&P_{1+1}=\dfrac{5-1}{1+1} P_1\\ \implies &P_2=2\cdot 5=10.\end{align} For $r=2$ \begin{align}&P_{2+1}=\dfrac{5-2}{2+1} P_2\\ \implies
- Question 3 & 4 Exercise 4.3
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... tan. =====Question 3===== Find sum of all the numbers divisible by $5$ from $25$ to $350$. GOOD ====Solution==== The numbers divisible by $5$ from $25$ tò $350$ are\\ $$25,... , d=5$ and $a_n=350$\\ To find $n$, we know that \begin{align}a_n&=a_1+(n-1) d\end{align} in the given
- Question 5 & 6 Exercise 4.3
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... war, Pakistan. =====Question 5===== Find four numbers in an arithmetic sequence, whose sum is $20$ an... uares is $120$ . ====Solution==== Let the four numbers are\\ $$a-2 d, a-d, a+d, a+2 d,$$ $Condition-1$\\ Their sum is $20$ , thus\\ \begin{align}a-3 d+a-d+a+d+a+3 d&=20 \\ \Rightarrow 4
- Question 14 Exercise 4.2
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... Question 14(i)===== Insert three arithmetic means between 6 and 41. GOOD ====Solution==== Let $A_1, A_2, A_3$ be three arithmetic means between 6 and 41. Then $6, A_1, A_2, A_3, 41$ are in A.P. We have $$a_1=6 \tex
- Question 15 & 16 Exercise 4.5
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... amount each day. If this continụed, how much must be set aside on the $15^{\text {th }}$ day? What is ... ves on $15^{1 / 2}$ day is $$a_{15}=a_1 r^{14} $$ becomes in the given case $$a_{15}=1 .(2)^{1 4}=R s... 30}-1)}{r-1} $$ putting $r-2$ and $a_1=1$, then \begin{align}S_{30}&=\dfrac{1[2^{30}-1]}{2-1}=2^{30}-
- Question 12 & 13 Exercise 4.2
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... f A.P and we have to find $a_{21}$. As, we have \begin{align} a_{21}&=a_1+20d\\ &=3500+20(750) \\ &=1... =====Question 13(i)===== Find the arithmetic mean between $12$ and $18$. GOOD ====Solution==== Here $a=12, b=18$.\\ Let say $A$ be arithmetic means. Then \\ \begin{align}A&=\dfrac{
- Question 1 Exercise 4.5
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... of series.\\ We know that $$a_n=a_1 r^{n-1}$$,\\ \begin{align}3.2^9&=3(2)^{n-1} \text { or }(2)^{n-1}=... w }\quad S_n&=\dfrac{a_1(r^n-1)}{r-1},\end{align} becomes in the given case\\ \begin{align}S_{10}&=\dfrac{3[2^{10}-1]}{2-1} \\ \Rightarrow \quad S_{10}&=3
- Question 16 Exercise 4.2
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... ====Question 16===== Insert five arithmetic means between $5$ and $8$ and show that their sum is five times the arithmetic mean between $5$ and $8$. GOOD ====Solution==== Let $A_1, A_2, A_3, A_4, A_5$ be five arithmetic means between $5$ and $8$. Then $
- Question 1 Exercise 4.3
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... = Find indicated term and sum of the indicated number of terms in arithmetic sequence: $9,7,5,3, \ldot... h term; 20 terms. GOOD ====Solution==== Let $a_1$ be first term and $d$ be common difference of given A.P. Then \begin{align}&a_1=9 \\ &d=7-9=-2 \\ &n=20
- Question 10 Exercise 4.4
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... war, Pakistan. =====Question 10===== Find two numbers if the difference between them is $48$ and their A.M exceeds their G.M by $18$ . ====Solution==== Let the two numbers be $a$ and $b$ \\ Condition-$1$\\ The differenc
- Question 3 and 4 Exercise 4.1
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... sequence to pick the pattern of the sequence as: \begin{align} &(-1)^2 \cdot 2 \cdot 1, (-1)^3 \cdot 2... sequence to pick the pattern of the sequence as: \begin{align}(-1)^2,(-1)^3,(-1)^4,(-1)^5, \ldots, (-1... on==== Given $$a_1=3, a_{n+1}=5-a_n.$$ For $n=1$ \begin{align}a_{1+1}&=5-a_1\\ \Rightarrow a_2&=5-3=2
- Question 9 & 10 Exercise 4.3
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... estion 9===== Find the sum 'of all multiples of 9 between 300 and 700. ====Solution==== All the multiples of 9 between 300 and 700 are:\\ $$306,315,324,333, \ldots... 5-306) = 9 \text { and } a_n=693 .$$\\ Let the number of terms be $n$. Then\\ \begin{align}a_n&=a_1+(n
- Question 13 & 14 Exercise 4.3
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... seats are there 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 s... rithmetic equence.\\ We have to find the total number of seats that are $S_{40}$.\\ We know by sum for
- Question 11 Exercise 4.4
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... that the prodect of $\mathrm{n}$ geometric means between $a$ and $b$ is equal to the $nth$ power for the single geometric mean between them. ====Solution==== Let $G_1, G_2, G_9, \ldots, G_n$ be the $n$ geometric means between $a$ and $b$,\\ th
- Question 9 Exercise 4.4
- ok of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Pesha... ===Question 9(i)===== Insert five geometric means between $3 \dfrac{5}{9}=\dfrac{32}{9}\quad$ and $\qu... ==Solution==== Let $G_1, G_2, G_3, G_4$ and $G_5$ be the five geometric means between $\dfrac{32}{9}$ and $\dfrac{81}{2}$,\\ then $\dfrac{32}{9}, G_1, G_2