Induction Proof again :(
Once again I'm pretty stumped at this one.
3^3 = 27
3^4 = 81
3^6 = 729
3^7 = 2187
Now these are the same practically. An even second to last digit and a seven as the last digit. So the pattern will loop. Just proving it seems a right pain! :( Any help at all would be appreciated.
I can't see your linked image, and I doubt anyone else can either.
Originally Posted by Jimbobobo
Prove by induction that in the decimal form 3^n, the second from end digit is even.
I have a slight variation of RonL's solution.
We note that ends in: 1, 3, 7, or 9.
Prove by induction that in the decimal form of , the tens-digit is even.
And ends in: 03, 09, 21, or 27.
. . Hence, there is a "carry" of either 0 or 2.
Verify . . . true!
Assume has an even tens-digit.
. . This means that: . is of the form: .
. . Hence: .
Multiply by 3: .
Therefore, has an even tens-digit.
. . The inductive proof is complete.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
I had this wild idea of using binary, but I got nowhere.
I thought that . would lead to a neat solution,
. . but I didn't find one.
Anyone care to give it a try?