Proof by Induction with Exponents

[Texans80mvp]

Homework Statement

By mathematical induction, prove that for n ≥ 1, 4/(7ⁿ - 3ⁿ).

Homework Equations

The Attempt at a Solution

I got the base case down P(1): 7-3=4.

Now the actual problem,

7ⁿ - 3ⁿ = 4x
7ⁿ⁺¹ - 3ⁿ⁺¹ = 7(7ⁿ) - 3(3ⁿ)
=7(4x + 3ⁿ) - 3(7ⁿ - 4x)
=21x+ (7(3ⁿ)) - (3(7ⁿ)) + 12x

-This is the point at which I get stuck there is nothing I can really factor out and I'm pretty sure I messed up earlier or there is something I have to do with the 7 and 3ⁿ. Any help would be appreciated.

Trying to get to: 7ⁿ⁺¹ - 3ⁿ⁺¹

 

[Infrared]

Here is how I would go about it. Since we care about 7^n and 3^n mod 4, it might be helpful to write 7^n=4a+r and 3^n=4b+r where r is the remainder when 7^n (or 3^n) is divided by 4 (Why must r be the same for both expressions?)
Write 7^(n+1)=7*7^n and 3^(n+1)=3*3^n and substitute.

 

[LCKurtz]

Homework Statement

By mathematical induction, prove that for n ≥ 1, 4/(7ⁿ - 3ⁿ).

Homework Equations

The Attempt at a Solution

I got the base case down P(1): 7-3=4.

Now the actual problem,

7ⁿ - 3ⁿ = 4x
7ⁿ⁺¹ - 3ⁿ⁺¹ = 7(7ⁿ) - 3(3ⁿ)
=7(4x + 3ⁿ) - 3(7ⁿ - 4x)

Try not substituting that second term so you have$$
7(4x+3^n) - 3\cdot 3^n$$and see if you can factor out a 4 from that.

 

[Dick]

Homework Statement

By mathematical induction, prove that for n ≥ 1, 4/(7ⁿ - 3ⁿ).

You probably meant to write 4|(7^n-3^n). Not '/'. The '|' means 'divides'.