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I keep getting questions like: $$ \frac 12+\frac34+\frac56+...+\frac{2n+1}{2n+2}>\frac {1}{\sqrt{3n+4}}$$

And I understand the method of setting it up but I cannot grasp the concept of fake math when I say $x>y>z => x>z $ How am I supposed to know what z should be without flat out saying k+1?

Please help me understand this before I get killed with a test.

If it's something basic such as x>y>z it's easy, still not real math as it has no equal signs but I'm able to understand. It's the roots that throw me off. Look at the equation that I posted that is the type of problem that I have trouble showing work on. It's obvious k

I would obviously be failed if I said k

YiFan Tey
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Karl
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    If you know that you are older than Sam and Sam is older than Tom , Then you can easily understand that you are older than both Sam and Tom. Similarly if you we know that $x > y $ and $y > z$ then $x > z$.
    Also the first principle of mathematical Induction states that if the equation is true for $P(n)$ and you are able to show that it is true for $P(K+1)$ if $P(K)$ is True , Then it implies it must be true for all possible values of $n$     – The Demonix _ Hermit Oct 02 '19 at 05:55
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    What do you mean by "I cannot grasp the concept of fake math"? – almagest Oct 02 '19 at 08:46
  • This seems to be a trend with your posts: distracting and unnecessary remarks unrelated to the question. For instance, comments like "still not real math as it has no equal signs", "It's obvious k", "get killed with a test" and addressing the users of this site as "useless plebians" and commanding us to "answer the question already". Please learn to observe basic courtesy, we are helping you and you are not entitled to a response. In the meantime, I have rolled back the edit which introduced the last unnecessary paragraph. – YiFan Tey Oct 06 '19 at 01:48

1 Answers1

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Suppose you have a proposition $P_n$ for each $n=1,2,3,4\dots$ and you know the following:

(1) $P_1$ is true

(2) if $P_1$ is true, then $P_2$ is true

(3) if $P_2$ is true, then $P_3$ is true

(4) if $P_3$ is true then $P_4$ is true

Then presumably you are quite happy to conclude that $P_4$ is true.

The principle of induction extends this idea by giving you the equivalent of infinitely many statements like (2), (3), (4) which are summarised in the single statement:

(*) if $P_k$ is true for any positive integer $k$, then it is also true for $P_{k+1}$.

Then the combination of (1) and (*) allows you to deduce that $P_n$ is true for any positive integer $n$.

almagest
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  • This has nothing to do with the equation I'm asking about or solving inductive step with roots. – Karl Oct 05 '19 at 00:39