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- Question 12 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... eshawar, Pakistan. =====Question 12(i)===== Show by mathematical induction that $\dfrac{5^{2 n}-1}{24... thbf{N}$, $\dfrac{5^{2 k}-1}{24} \in \mathbb{Z}$ by (a). Thus the given statement is true for $n=k+1$. Hence by mathematical induction it is true for all $n \in
- Question 13 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... gn} & 2^{k+1}=2^k \cdot 2>k \cdot 2 \quad \text { by (i) } \\ & \Rightarrow 2^{k+1}>2 k=k+k \\ &\Right... the form of proposition taken when $n$ is replace by $k+1$, hence true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$
- Question 7 & 8 Review Exercise 7
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... tive intege: n. prove that $7^n-3^n$ is divisible by 4 . Solution: We using mathematical induction to ... rrow 7^{k+1}-3^{k+1}=4.7^k+3.4 Q \end{aligned} $$ by induction hypothesis $$ \begin{aligned} & \Righta... \Rightarrow 7^{k-1}-3^{k+1} \text { is divisible by } 4 . \end{aligned} $$ Thus the given statement
- Question 1 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 1===== Establish the formulas below by mathematical induction, $2+4+6+\cdots+2 n=n(n+1)$... =(k+1)(k+1+1)\end{align} Which is the form taken by proposition when $n$ is replaced by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induct
- Question 2 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 2===== Establish the formulas below by mathematical induction, $1+5+9+\ldots+(4 n-3)=n(2... k+1)[2(k+1)-1]\end{align} Which is the form taken by proposition when $n$ is replaced by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induct
- Question 9 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 9===== Establish the formulas below by mathematical induction, $\dfrac{1}{3}+\dfrac{1}{9... ^{k-1}}\right]\end{align} Which is the form taken by proposition when $n$ is replaced by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induct
- Question 10 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... ====Question 10===== Establish the formulas below by mathematical induction, $\left(\begin{array}{1}5 ... ight)\end{align} Which is the just the form taken by given propusition when $n$ is replaced by $k+1$. hence it is true for $n=k+1$. Thus by mathematical
- Question 14 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... -1}+2^{2 k-1}) \\ & =5.3^{2 k}+5.4 . Q\end{align} by induction hypouresis $$=5 [3^{2 k-1}+4 \cdot Q]$$... Hence given statement is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in ... .2^{2 k}+2^{2 k}-1 \\ & =3.2^{2 k}+3 Q\end{align} by induction hypothesis $$=3[2^{2 k}+Q]$$ $3$ divide
- Question 3 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 3===== Establish the formulas below by mathematical induction $3+6+9+\ldots+3 n=\dfrac{3... the form of given statement when $n$ is replaced by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathb
- Question 4 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 4===== Establish the formulas below by mathematical induction $3+7+11+\cdots+(4 n-1)=n(2... the form of given statement when $n$ is replaced by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathb
- Question 5 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 5===== Establish the formulas below by mathematical induction, $1^3+2^3+3^3+\ldots+n^3=\... he form taken ny proposition when $n$ is replaced by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathb
- Question 6 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 6===== Establish the formulas below by mathematical induction, $1(1 !)+2(2 !)+3(3 !)+\ld... hich is the form proposition when $n$ is replaced by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathb
- Question 7 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 7===== Establish the formulas below by mathematical induction, $1.2+2.3+3.4+\ldots+n(n+1... the form of the proposition when $n$ is replaced by $k+1$, hence it is true for $n=k-1$. Thus by mathematical induction it is true for all $n \in \mathb
- Question 8 Exercise 7.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... =====Question 8===== Establish the formulas below by mathematical induction, $1+2+2^2+2^3+\ldots+2^n 1... the form of the proposition when $n$ is replaced by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all positive inte
- Question 1 Exercise 7.2
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... shawar, Pakistan. =====Question 1(i)===== Expand by using Binomial theorem: $(x^2-\dfrac{1}{y})^4$ ==... y^4} \end{align} =====Question 1(ii)===== Expand by using Binomial theorem: $(1+x y)^7$ ====Solution=... . } \end{align} =====Question 1(iii)===== Expand by using Binomial theorem: $(\sqrt{y}+\dfrac{1}{\sqr