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'Diverging' printed from https://nrich.maths.org/
The first part of the question asks you to show that for natural
numbers $x$ and $y$ if ${x\over y}> 1$ then
$${x\over y}> {(x+1)\over(y+1)}> 1.$$
Here's a hint for this: try starting with $x> y$, which you are
given, and adding $xy$ to both sides of the inequality.
For the next part of the question you are given a product $P$ and
the hint to consider $P^2$ and clearly the first part of the
question should come in useful. Look out for a 'magic concertina'
effect!!
If you can prove the second inequality then you will have shown
that $P$ gets bigger and bigger without limit as you put more terms
into the product which proves that the product diverges, hence the
title of the question!
For the last bit of the question, taking $k=100$ and repeating the
last trick leads to the disappointing conclusion that $Q^2>
101$, this estimate is not good enough.
Go back to the drawing board and do some neat estimating of $Q^2$
calculating the product of the first few terms exactly and using
the concertina method on the rest. This will quickly give you the
result. This is a good illustration of what mathematicians do all
the time with inequalities. They go on sharpening them to get
better and better estimates until they get close enough for their
purposes.