So the first bit of the question gives you the approximation...
with and
...to find the approximate value of , which I've done.
The the second part is...
The Extended Mean Value Theorem says that for fixed , and , there is a in the range , such that
Use this to show that for some with we have
...which I think I've managed okay, but it's this next bit that I'm stuck on...
Deduce more generally that
...for some integers and numbers where .
Can anyone help? I just can't get my head around it.
I didn't mean to say that I should have known that you would not understand (whatever specific argument). Instead, what I wanted to say (but failed to communicate clearly) is that I should have known from your original question that you were primarily interested in getting some help for the third part (marked red).
Therefore it was not you who failed (to understand) but me who failed to read your question correctly.