# Higher IB induction proof - what am I missing??

Discussion in 'Mathematics' started by beedge, Dec 21, 2011.

1. ### beedgeNew commenter

I've been brushing up my all-round skills since we broke up, and I'm come across a question I just cannot do; namely to prove (using induction) that
7^n - 4^n - 3^n is divisible by 12
It starts off easily enough:
7^k - 4^k - 3^k = 12A
Then considering:
7^k+1 - 4^k+1 - 3^k+1, which of course is:
7(7^k) - 4(4^k) - 3(3^k) If you substitute for 7^k, you end up with:
84A + 3(4^k) + 4(3^k) which is where I'm stuck.
Ahhh...... The process of typing my thoughts has helped me to spot something I haven't spotted the last half an hour. Can you start converting the 3(4^k) into a 12(4^k-1)? Is that allowed?

2. ### beedgeNew commenter

I've been brushing up my all-round skills since we broke up, and I'm come across a question I just cannot do; namely to prove (using induction) that
7^n - 4^n - 3^n is divisible by 12
It starts off easily enough:
7^k - 4^k - 3^k = 12A
Then considering:
7^k+1 - 4^k+1 - 3^k+1, which of course is:
7(7^k) - 4(4^k) - 3(3^k) If you substitute for 7^k, you end up with:
84A + 3(4^k) + 4(3^k) which is where I'm stuck.
Ahhh...... The process of typing my thoughts has helped me to spot something I haven't spotted the last half an hour. Can you start converting the 3(4^k) into a 12(4^k-1)? Is that allowed?

3. ### Null

7*7^n-4*4^n-3*3^n = 7 (7^n-4^n-3^n) + ......+ ........

4. ### Polecat

Quite a sneaky one for IB.
Is this an actual IB question, or just found in a textbook.

5. ### Null

I was about to go out as I made my last post. Now that I look properly at your post I see that you had in fact solved the problem.

If you want another, you might like this one.

Prove that 5^(2n+2)-24n-25 is divisible by 576 for n=1,2,3.....

6. ### beedgeNew commenter

yes in a text book - definitely harder than typical divisibility induction questions

7. ### beedgeNew commenter

Null,
I'm not sure what you were doing in your first post. However, from you second post, does that mean that the following is correct:
84A + 3(4^k) + 4(3^k)
= 84A + 12(4^k-1) + 12(3^k-1)
= 12 (7A + 4^k-1 + 3^k-1)
And that's it?
I will do my best that other problem you set. Could take a while...

8. ### Colleen_YoungOccasional commenter

beedge that's what I got - and so did some other chap on a forum I found later!

9. ### Null

7*7^n-4*4^n-3*3^n = 7 (7^n-4^n-3^n) + ......+ ........

was short for

7*7^n-4*4^n-3*3^n = 7 (7^n-4^n-3^n) + 3*4^n + 4*3^n.

patronisingly (?) leaving the gaps for you to fill in.

As you see it was what you'd already done!

10. ### beedgeNew commenter

Prove that 5^(2n+2)-24n-25 is divisible by 576 for n=1,2,3.....
Did it! What an elegant little proof. My first thought was "this looks horendous" but it worked out very nicely. Do you have any other good ones?