- #1
skiing4free
- 20
- 0
Homework Statement
Prove the following statement by induction
5|(7^k-2^k) for all k[tex]\in[/tex]R
2. The attempt at a solution
Started by proving P(k)=5|(7^k-2^k) where k=0
which gives 5|0=7^0-2^0 =>0
Then when P(k+1)=5|(7^k-2^k) it gets more complicated and I get stuck.
Proving this induction: 7^(k+1)-2^(k+1)=7.7^k-2.2^k
But this is as far as a get and I cannot seem to get the nest step.
The answers show that the next steps are:
=> 7.7^k-7.2^k+7.2^k-2.2^k
=> 7(7^k-2^k)+(7-2)2^k
=> 5(7x+2^k)
Which apparently is the proof.
I'm confused as to how they get from the 1st step to the second and also where the variable x comes from in the final step.
Any help on this question would be greatly appreciated.