What do we actually prove using induction theorem? Here is the picture of the page of the book, I am reading:


$$P_k: \qquad 1+3+5+\dots+(2k-1)=k^2$$
  Now we want to show that this assumption implies that $P_{k+1}$ is also a true statement:
  $$P_{k+1}: \qquad 1+3+5+\dots+(2k-1)+(2k+1)=(k+1)^2.$$
  Since we have assumed that $P_k$ is true, we can perform operations on this equation.
  Note that the left side of $P_{k+1}$ is the left side of $P_k$ plus $(2k+1)$.
  So we start by adding $(2k+1)$ to both sides of $P_k$:
  \begin{align*}
1+3+\dots+(2k-1) &= k^2  &P_k\\
1+3+\dots+(2k-1)+(2k+1) &= k^2+(2k+1) &\text{Add $(2k+1)$ to both sides.}
\end{align*}
  Factoring the right side of this equation, we have
  $$1+3+\dots+(2k-1)+(2k+1) =(k+1)^2 \qquad P_{k+1}$$
  But this last equation is $P_{k+1}$. 
  Thus, we have started with $P_k$, the statement we assumed true, and performed valid operations to produce $P_{k+1}$, the statement we want to be true. 
  In other words, we have shown that if $P_k$ is true, then $P_{k+1}$ is also true. 
  Since both conditions in Theorem 1 are satisfied, $P_n$ is true for all positive integers $n$.

In the top line there is written that we have to show that 'the assumption $P_k$ is true implies $P_{k+1}$ is true'.
And what I think is that: as long as we know that the state of a proposition being true for any positive integer $k$ after Base number implies that proposition is true for integer $k+1$, we have to show that $P_k$ is true. However I don't have an idea yet how to show the truth of $P_k$.
So, my first question is that who is right? My book or me? And if I am right then how can I show the truth of $P_k$?
2) This may be taken as the second question but it is also bit annoying.
In the second paragraph it is written, "Since we have assumed that $P_k$ is true, we can perform operations on it". 
Why is it necessary for an equation to be true for performing operations on it? Well, this is not much annoying as compared to the first question because whole the induction theory depends upon that. 

Induction is, intuitively, an outline of an infinite proof.
You first prove $P(0)$, the base case.
Then you prove $P(1)$ follows from $P(0)$.
Then you prove that $P(2)$ follows from $P(1)$.
Et cetera. 
In general, if you know that $P(k+1)$ follows from $P(k)$, and you know that $P(0)$ is true, then you know how to prove $P(n)$ for any natural number. For example, $P(4)$ follow because $P(0)\implies P(1)\implies P(2)\implies P(3)\implies P(4)$.
So you don't need to know how to show $P(k)$ to show that $P(k)$ implies $P(k+1)$. Once you've shown this implication, and $P(0)$, we know how to write a non-inductive finite proof of $P(10000)$ or $P(10^{99})$.
