I am kind of confused on how to solve this question. Can anyone help please

Question

If we let $S$ be the set that is defined by the following two rules:

• 1 is an element of the set $s$
• If $s$ is an element of the set $s$, then x+$2 \sqrt{x}+1$ is also an element of the set $s$ how can I show and prove the simple discription of the set $s$

Answer 1

Well, start by plugging in the first several elements of the set and looking for a pattern.

So, we have 1 is an element of the set. Thus, by the second rule,

$1+2\sqrt 1+1=4$ is an element of the set Using 4, we get; $4+2\sqrt 4 +1=9$ is an element of the set

Using 9, we get $9+2\sqrt 9 +1=16$ is an element of the set.

Do you see the pattern?

Another way of looking at it is to look at $x+2\sqrt x +1$ as the 'multplied out' version of the perfect square $(\sqrt x +1)^2$. This makes it even more clear that each element you are getting is going to be a square, and you should be able to see that you hit every square.