With Mathematics Induction, Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11` for all natural number `n`

 With Mathematics Induction, Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11` for all natural number `n` .

Solution

Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11`

• If `n=0` then `A_0=2+9=11` It is true that `A_0 \equiv0(mod11)`
• If `n=1` then `A_1=2^7+9^2=209=11.19` It is true that `A_1 \equiv0(mod11)`

We assuming that: `A_k \equiv0(mod11)` for all natural number `k`

We will prove that It is true till `n=k+1` then`A_(k+1) \equiv0(mod11)`?

From the value of `A_(k+1)=2^(6n+7)+9^(n+2)`

                               `A_(k+1)=2^6(2^(6k+1)+9^(k+1))+(9^(k+2)-2^6.9^(k+1))`

                                `A_(k+1)=64A_k+9^(k+1)(9-64)`

                                `A_(k+1)=64A_k-55.9^(k+1)`

From our assuming: `A_k \equiv0(mod11)` then It is easy to see that `A_(k+1) \equiv0(mod11)`

Hence, The problem is proved.

Solution by: Thin Sokkean

Practice Problem For You

1- It is given that : `E_n=831^n+709^n-743^n-610^n` for all natural number `n`.

    Prove that: `E_n` is divided by `189` for all natural number `n`.

    Hint: Using modulo formula and `gcd(9,21)=189`

2- Prove that for all natural number `n` we have the following inequation:
           `1+1/\sqrt{2}+1/\sqrt{3}+....+1/sqrt{n+1}<2sqrt{n+1}` 

    Hint: Using Mathematics Induction.

3- It is given natural real sequence satisfied that: 

                    `U_0=\sqrt{2}` and `U_(n+1)=sqrt{2+U_n}`

    a. Find the `U_n` which is functioned to `n`.

    b. Find the product of `P_n=U_0.U_1.U_2......U_n`

4- There is a 4 digits number with every single digit is `a` ; `a` ; `b` ; `b` correct order. 

    Find those number If It is a perfect square.

5- It is given that : `33^2=1089` , `333^2=110889` , `3333^2=11108889` , `33333^2=1111088889`.

    From the following given, let find the general term of it and prove.

6- a. Prove that: `1+1/cosx=(cotx/2)/(cotx)`.

    b. Calculate the product of :
    `P_n=(1+1/cosa)(1+1/cos(a/2))(1+1/cos(a/2^2))......(1+1/cos(a/2^n))`.

7- Calculating the value of : `S=cos^3(\pi/9)-cos^3(4\pi/9)+cos^3(7\pi/9)`.

8- Calculating the sum of : `S_n=9+99+999+....+999` which is `n` times of number `9`.

9- Find all pair of integer `(m,n)>2`satisfied that: for all positive integer 

                    `a`  we got `(a^m+a-1)/(a^n+a^2-1)` be a integer. 

        Solution is `(m,n)=(5,3)`

10- It is given `3` positive integer `a,b,c` satisfied that `a+b+c=10`.

       Find the minimum value of `P=a\timesb\timesc`.

        Solution is `P=36`.

11- Find the exact value of `sin(\pi/10)` and `cos(\pi/10)`.

12- It is given two positive real number `a` and `b`. Prove that: `(1+a)(1+b)\geq(1+\sqrt{ab})^2`

       From the proven, let's minimum of function :
                `f(x)=(1+4^(sin^2x))(1+4^(cos^2x))` for all  real number `x`.

13- It is given three real numbers `a,b,c` .

        Prove that: `a^2+b^2+c^2` ≥ `ab+bc+ac`

14- It is given `n` positive real numbers `a_1;a_2;a_3;.....;a_n` satisfied that
        `a_1.a_2.a_3......a_n=1`.

       Prove that:  `(1+a_1)(1+a_2)(a+a_3)........(1+a_n)` ≥ `2^n`.

15- It is given `m,n` be the positive integers . Prove that for all real positive number `x` 
             `(x^(mn)-1)/m` ≥`(x^n-1)/x`

16- For all real number `x` prove that: 

            `(1+sinx)(1+cosx)` ≤ `3/2+\sqrt{2}`

17- It is given `x_n=2^(2^n)+1` for `n=1,2,3......`

       Prove that: `1/x_1+2/x_2+2^3/x_3+......+2^(n-1)/x_n` < `1/3`.

18- It is given function `y=(x^2+2mx+3m-8)/(2(x^2+1)` which `x` is real number and `m` is a 

       parameter of the problem. Is It possible that we can find the value of `m` to make 

       function `y` be the value of `cos` of one single angle ?

19- It is given two variables function:
            `f(x,y)=((x^2-y^2)(1-x^2y^2))/((1+x^2)^2(1+y^2)^2) ;x,y` be real number.

       Prove that: `|f(x,y)| \leq1/4`.

20- It is given `\theta` be a real number such that `0<\theta<\pi/2`.

       Prove that: `(sin\theta)^(cos\theta)+(cos\theta)^(sin\theta)>1`.

21- There are three real number: `a>0 ; b> ; c>0` Prove that:
        `ab(a+b)+bc(b+c)+ac(a+c)\geq6abc`

22- Solve the equation below:
       `9^((x^2-x))+3^((1-x^2))=3^((x-1)^2)+1`

23- Let's finding both functions `f(x)` and `g(x)` which satisfied that: 

               `f(2x-1)+2g(3x+1)=x^2`

                `f(4x-3)-g(6x-2)=-2x^2+2x+1`

23- Find the sum of:
       `S_n=tana+1/2tan(a/2)+1/2^2tan(a/2^2)+....+1/2^ntan(a/2^n)`

24- It is `3rd` degree polynomial `P(x)` which is satisfied that:
                    `P(x)+2` is divided by `(x+1)^2`

                    `P(x)-2` is divided by `(x-1)^2`

      Determind the polynomial `P(x)` ?

25- Let `A=(1/\sqrt{3}+i)^n-(1/\sqrt{3}-i)^n` for all natural number `n`.

       Prove tha: `A=i.2^(n+1)/(\sqrt{3})^n.sin(n\pi/3)`.

26- Solve the equation in integer set :

                    `47x+29y=1`

27- Find all function `f(x)` possible which is satisfied that:
                    `f(x+\sqrt{x^2-2x+1})=(x^2-1)/(x^2+1)`

28- 

Happy Learning!