Here is the basic trick.
I need to prove 11^n - 6 is divisible by 5 for all n >or = to 1 by induction.
So I start by basis step 11^1 -6 get 5 and 5 is divisible by 5 so basis step is true.
Then I am suppose to add n +1 to both sides and show it is still true.
I am stumped!!! Anyone willing to help a very old and confused student???!!!
You've got the base case so I'll continue from there.
Let be divisible by 5 for some positive integer k. Then we may define:
where x is some positive integer.
We wish to show that is divisible by 5. So:
Now recall that
Since x is an integer, so is 11x + 12. Thus etc.
Another approach . . .
Prove by induction: . is divisible by 5 for all
Verify . . . True!
Assume for some integer
Add to both sides: .**
Therefore: . is divisible by 5.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
How did I know what to add? . . . Simple!
I want: .
. . Then: .
Therefore: . . . . . see?