Show that is divisible by 7 for all positive integers

Thanks for your time guys, really appreciate it and sorry about the dodgy math after the 92, I'm not very good when it comes to this computer stuff.

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- Sep 22nd 2007, 02:39 PMBartimaeusShow that it is divisible by 7 for all positive integers...
Show that is divisible by 7 for all positive integers

Thanks for your time guys, really appreciate it and sorry about the dodgy math after the 92, I'm not very good when it comes to this computer stuff. - Sep 22nd 2007, 02:51 PMtopsquark
- Sep 22nd 2007, 03:03 PMtopsquark
If you don't know modular arithmetic, then consider n = 1:

which is divisible by 7.

Now assume the theorem to be true for some n = k.

Assume that

where x is some integer.

Consider the case for n = k + 1:

This should be divisible by 7.

Now, consider again:

Solving this for gives:

Inserting this into the k + 1 expression gives:

The last term is manifestly divisible by 7. Let's look at the first term:

The coefficient of the is

So

where x is an integer. This is obviously divisible by 7.

Thus is divisible by 7 for n = 1, thus for n = 2, thus etc.

-Dan