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
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 $n$.