Alright, here is the problem:
using the principle of mathematical induction prove that, for any positive integer n,
n^4 + (n+1)^4 + (n+2)^4 + (n+3)^4 + 14 is divisible by 16.
What I have done so far is the base step, and I have assumed the statement is true for n=k.
For n=k+1 I have added (k+4)^4 - k^4 to the expression for n=k. Once I came up with that equation, I expanded all the parts...
I ended up with 4k^4 + 40k^3 + 180k^2 + 400k + 368
now I have no idea where to go with it...
thank-you in advance for any help.
Use your inductive assumption.
You want to see if the statement is true for .
Well you seem to have correctly observed that the expression for is just the expression for with added on.
Since by inductive hypothesis, 16 divides the expression for , all you have to do is verify that 16 divides .
Expand this and you will see every term is divisible by 16. (you may find it easy to expand if you take advantage of its form as a difference of squares).
In your proof, you might want to explain that if a number (16) divides two numbers, then it divides their sum as well. Presumably, this property has been discussed/proved in your class or textbook by now, so just mentioning the fact will probably suffice.
(Chuckles) Well, that was a lot easier than mine! :)
Originally Posted by Soltras
Yours is much better actual analysis of what's really going on though, and it is also providing me with a great modular arithmetic brain refresh which I badly need. :D
Originally Posted by topsquark
Your game plan is excellent . . . you're that close to the punch line.
We assume the statement is true for
. . for some integer
Add to both sides:
Therefore, we have proved