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Think I have parts 1 and 2, struggling with the induction :S Any help greatly appreciated!
There is no need for induction on this question when you can prove it simply by re-writing the inequality but very well...
I am going to assume there was a misprint and they omitted that the result holds true for .
OP, from your induction hypothesis, you have for and you wish to show with the assumption that .
Now, recall that .
Multiply the inequality through by and the result falls out fairly quickly with some manipulation.
@Plato, I think you mean for as equality holds for .
Also, this may be pedantic but I think you could make it a bit clearer that what you've done is that you've taken your inductive assumption and you've multiplied that through by and then added 1.
Clearly if .
Substitute for and then you have the rightmost inequality in my above post. You will be able to simplify that down into a fraction, call it . The denominator of will clearly be and so clearly iff the numerator of . Try it, you'll be able to sneak in the result from part C.
Also, I see you're a fellow Scot! Are you studying Mathematics @ Glasgow?