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- Question 1 Exercise 7.1
- Question 2 Exercise 7.1
- Question 3 Exercise 7.1
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- Question 5 Exercise 7.1
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- Question 7 Exercise 7.1
- Question 8 Exercise 7.1
- Question 9 Exercise 7.1
- Question 10 Exercise 7.1
- Question 11 Exercise 7.1
- Question 12 Exercise 7.1
- Question 13 Exercise 7.1
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- Question 15 Exercise 7.1
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- Question 2 Exercise 7.2
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- Question 11 Exercise 7.2
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- Question 1 Review Exercise 7
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- Question 12 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... awar, Pakistan. =====Question 12(i)===== Show by mathematical induction that $\dfrac{5^{2 n}-1}{24}$ is... }{24}=\dfrac{5^{2.1}-1}{24}=\dfrac{24}{24}=1 \in \mathbb{Z}$$ Thus it is true for $n=1$ 2. Let it be true for $n=k>1$ then $$\dfrac{5^{2 k}-1}{24} \in \mathbb{Z}$$ 3. For $n=k+1$ then consider \begin{align}
- Question 11 Exercise 7.2
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... $\left(\begin{array}{l}n \\ r\end{array}\right)=\mathrm{C}_r$. Show that $\mathrm{C}_1+2 \mathrm{C}_2 x+3 \mathrm{C}_3 x^2+\ldots \ldots . .+\mathrm{nC}_{\mathrm{n}} x^{\mathrm{n}-1}=
- Question 14 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ence given statement is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. =====Question 14(ii)===== Prove that $2^{2 n}... ^{2 k}-1$ or $2^{2 k}-1=3 Q$ where $Q \subseteq \mathbb{Z}$ is quotient. 3. For $n=k+1$ then we have \
- Question 13 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... . =====Question 13(i)===== $2^n>n \forall n \in \mathbf{N}$. ====Solution==== 1. For $n=1$ then $2^n=2^... place by $k+1$, hence true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. =====Question 13(ii)===== $n$ ! $>n^2$ for eve
- Question 1 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 1===== Establish the formulas below by mathematical induction, $2+4+6+\cdots+2 n=n(n+1)$ ====... )^{t h}$ term of the series, which is: $$a_{k+1}=\mathbf{2}(k+1)=2 k+2 $$ Adding this $k+1$ term to both... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it it true for $n \in \mathbf{N
- Question 2 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 2===== Establish the formulas below by mathematical induction, $1+5+9+\ldots+(4 n-3)=n(2 n-1)... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it it true for $n \in \mathbf{N}$. ====Go To==== <text align="left"><btn type="p
- Question 3 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 3===== Establish the formulas below by mathematical induction $3+6+9+\ldots+3 n=\dfrac{3 n(n+... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ====Go To==== <text align="left"><btn ty
- Question 4 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 4===== Establish the formulas below by mathematical induction $3+7+11+\cdots+(4 n-1)=n(2 n+1)... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ====Go To==== <text align="left"><btn ty
- Question 5 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 5===== Establish the formulas below by mathematical induction, $1^3+2^3+3^3+\ldots+n^3=\left[... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ====Go To==== <text align="left"><btn t
- Question 6 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 6===== Establish the formulas below by mathematical induction, $1(1 !)+2(2 !)+3(3 !)+\ldots+n... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ====Go To==== <text align="left"><btn t
- Question 7 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 7===== Establish the formulas below by mathematical induction, $1.2+2.3+3.4+\ldots+n(n+1)=\df... by $k+1$, hence it is true for $n=k-1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ====Go To==== <text align="left"><btn typ
- Question 9 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 9===== Establish the formulas below by mathematical induction, $\dfrac{1}{3}+\dfrac{1}{9}+\df... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \leq \mathbf{N}$. ====Go To==== <text align="left"><btn ty
- Question 10 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... =Question 10===== Establish the formulas below by mathematical induction, $\left(\begin{array}{1}5 \\5 \... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ====Go To==== <text align="left"><btn type="
- Question 8 Exercise 7.1
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... ==Question 8===== Establish the formulas below by mathematical induction, $1+2+2^2+2^3+\ldots+2^n 1=2^n-... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all positive int... o To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit07:ex7-1-p7 |< Question 7 ]]</btn
- Question 7 & 8 Review Exercise 7
- ion and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtu... $7^n-3^n$ is divisible by 4 . Solution: We using mathematical induction to prove the given statement. (... re $x>-1$. - Solution: We try to prove this using mathernatical induction. 1. For $n=1$ then $$ (1+x)^1=... $$ Hence the given is true for $n=k+1$. Thus by mathematical induction the given is true for all $n \g