SOLUTION: 6x7^n - 2x3^n habis dibagi 4 untuk setiap n bilangan asli. keterangan : ^n = pangkat n

Algebra.Com
Question 1203186: 6x7^n - 2x3^n habis dibagi 4 untuk setiap n bilangan asli.
keterangan : ^n = pangkat n
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Translation of what the student wrote
6x7^n — 2x3^n is divisible by 4 for every n natural numbers.
description: ^n = power n

I'll write the solution in English, and let the student translate back to their original native language.

-----------------------------

I'm assuming that you want to prove this claim by induction.

The natural numbers is the set {1,2,3,4,...}
n = 1 is the smallest item
Compute the expression for n = 1
6*7^n - 2*3^n
6*7^1 - 2*3^1
6*7 - 2*3
42 - 6
36
4*9
The last step shows that we get a multiple of 4
Which in turn proves that 6*7^n - 2*3^n is a multiple of 4 when n = 1
In other words, 6*7^n - 2*3^n is divisible by 4 when n = 1

That covers the base case.
Now onto the inductive step.
Assume that 6*7^k - 2*3^k is a multiple of 4, when k is some natural number and k > 1.

This would mean
6*7^k - 2*3^k = 4m
for some integer m

Based on the assumption above, we want to prove that 6*7^(k+1) - 2*3^(k+1) is also a multiple of 4
The goal is to show that 6*7^(k+1) - 2*3^(k+1) can be written in the template 4*(some integer).

Through a bit of algebra...
6*7^(k+1) - 2*3^(k+1)
6*7^k*7^1 - 2*3^k*3^1
42*7^k - 6*3^k
(6*7^k + 36*7^k) - (2*3^k + 4*3^k)
6*7^k + 36*7^k - 2*3^k - 4*3^k
(6*7^k - 2*3^k) + (36*7^k - 4*3^k)
(4m) + 4*(9*7^k - 1*3^k)
4( m + 9*7^k - 1*3^k )
4*(some integer)

We conclude that 6*7^(k+1) - 2*3^(k+1) is a multiple of 4 when 6*7^k - 2*3^k is a multiple of 4.
This wraps up the inductive step, and it wraps up the induction proof overall.


---------------------------------------

Another approach to the inductive step

Return back to the assumption of
6*7^k - 2*3^k = 4m

Isolating the first term gets us
6*7^k = 4m + 2*3^k

Then apply a substitution as marked in red
6*7^(k+1) - 2*3^(k+1)
6*7^k*7^1 - 2*3^k*3^1
7*6*7^k - 6*3^k
7*6*7^k - 6*3^k
7*(4m + 2*3^k) - 6*3^k
28m + 14*3^k - 6*3^k
28m + 8*3^k
4*( 7m + 2*3^k )
4*(some integer)

We have shown that 6*7^(k+1) - 2*3^(k+1) is a multiple of 4 when 6*7^k - 2*3^k is a multiple of 4.

---------------------------------------

However you approach the inductive step, both methods show that 6*7^k - 2*3^k being a multiple of 4 leads 6*7^(k+1) - 2*3^(k+1) to be a multiple of 4.

It is then a domino effect to cause 6*7^n - 2*3^n to be a multiple of 4 for any natural number n.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Translation from Indonesian

6*7^n - 2*3^n is divisible by 4 for every natural number n.
description: ^n = power n
~~~~~~~~~~~~~~~~~~~~ 

First, notice that 7^n and 3^n give THE SAME REMAINDER when divided by 4.


    It is because the difference

        7^n - 3^n = (7-3) * (7^(n-1) + 7^(n-2)*3 + 7^(n-3)*3^2 + . . . 7^1*3^(n-2)  + 3^(n-1))

    is divizible by 7-3 = 4.  // Recall the general formula  

        a^n - b^n = (a-b) * (a^(n-1) + a^(n-2)*b + a^(n-3)*b^2 + . . . a^1*b^(n-2)  + b^(n-1)).



Therefore, 6*7^n — 2*3^n gives the same remainder, when divided by 4, as 6*3^n — 2*3^n, 

which simply equals to  (collect like terms)  4*3^n, 

which is, obviously, divisible by 4, since it is a multiple of 4.


At this point, the statement is proved and the problem is solved, in full.


------------------

There is a standard amount / (complex) of knowledge that advanced school student
should learn (of those students who attend Math schools, Math circles or read relevant Math literature).
This complex of knowledge is covered in lessons of this site, listed in overview lessons

    - OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11

    - OVERVIEW of miscellaneous solved problems on divisibility of integer numbers

Summer is good time to read these lessons,
so you have a rare and happy opportunity to make your knowledge wider and deeper.


//////////////////


Comment from student : Thanks this answer is very helpful.


My response : I am very glad.
Come again to this forum soon to learn something new.



RELATED QUESTIONS

N-(-4)n (answered by ikleyn)
-4 n( n+ 2) =... (answered by stanbon)
Simplify 3 4 5 -- + --- - --- n n... (answered by checkley71)
f(n)={n/2x3+1 if n is even , if n is odd (answered by ikleyn)
n+4 (answered by Alan3354)
{{{-2(n-8)>-n+4}}} (answered by bucky)
|n-(-4)|=6... (answered by vleith)
n+3 for... (answered by MathLover1)