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- Question 12 Exercise 7.1 @math-11-kpk:sol:unit07
- awar, Pakistan. =====Question 12(i)===== Show by mathematical induction that $\dfrac{5^{2 n}-1}{24}$ is an inte... the given statement is true for $n=k+1$. Hence by mathematical induction it is true for all $n \in \mathbf{N}$. =====Question 12(ii)===== Show by mathematical induction that $\dfrac{10^{n+1}-9 n-10}{81}$ is a... d$ by (i), hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$.
- Question 1 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 1===== Establish the formulas below by mathematical induction, $2+4+6+\cdots+2 n=n(n+1)$ ====Solution... 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 @math-11-kpk:sol:unit07
- ==Question 2===== Establish the formulas below by mathematical induction, $1+5+9+\ldots+(4 n-3)=n(2 n-1)$ ====So... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it it true for $n \in \mathbf{N}$.
- Question 3 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 3===== Establish the formulas below by mathematical induction $3+6+9+\ldots+3 n=\dfrac{3 n(n+1)}{2}$ ... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$.
- Question 4 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 4===== Establish the formulas below by mathematical induction $3+7+11+\cdots+(4 n-1)=n(2 n+1)$ ====So... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$.
- Question 5 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 5===== Establish the formulas below by mathematical induction, $1^3+2^3+3^3+\ldots+n^3=\left[\dfrac{n... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$.
- Question 6 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 6===== Establish the formulas below by mathematical induction, $1(1 !)+2(2 !)+3(3 !)+\ldots+n(n !)= -... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$.
- Question 7 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 7===== Establish the formulas below by mathematical induction, $1.2+2.3+3.4+\ldots+n(n+1)=\dfrac{n(n+... by $k+1$, hence it is true for $n=k-1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$.
- Question 8 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 8===== Establish the formulas below by mathematical induction, $1+2+2^2+2^3+\ldots+2^n 1=2^n-1$. ====... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all positive integers.
- Question 9 Exercise 7.1 @math-11-kpk:sol:unit07
- ==Question 9===== Establish the formulas below by mathematical induction, $\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{2... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \leq \mathbf{N}$.
- Question 10 Exercise 7.1 @math-11-kpk:sol:unit07
- =Question 10===== Establish the formulas below by mathematical induction, $\left(\begin{array}{1}5 \\5 \end{arra... by $k+1$. hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$.
- Question 13 Exercise 7.1 @math-11-kpk:sol:unit07
- place by $k+1$, hence true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ... by $k+1$, hence it is true for $n=k+1$. Thus by mathematical induction it is true for all $n \geq 4$. ====
- Question 14 Exercise 7.1 @math-11-kpk:sol:unit07
- ence given statement is true for $n=k+1$. Thus by mathematical induction it is true for all $n \in \mathbf{N}$. ... the given statement is true for $n=k+1$. Thus by mathematical induction the given statement is true for all $n
- Question 7 & 8 Review Exercise 7 @math-11-kpk:sol:unit07
- $7^n-3^n$ is divisible by 4 . Solution: We using mathematical induction to prove the given statement. (1.) For ... $$ Hence the given is true for $n=k+1$. Thus by mathematical induction the given is true for all $n \geq 1$.
- Unit 07: Mathmatical Induction and Binomial Theorem (Solutions) @math-11-kpk:sol
- s will be able to * Describe the principle of mathematical induction. * Apply the principle to prove the s