Big O analysis trying to follow a logic

Question

Can someone please help me understand why(the derivation) "and m = 2n+1 for each n."? I am trying to follow the logic of the solution provide while myself have a different approach. Here is my thinking (questions state below):

The inner for loop runs m times from 1 and the outer for loop runs n times from 1. Moreover, for each inner loop, m is increased by 2 based on previously additions and m starts at 1. With all these observations, plug in n = 1, print (j) get executed 3 times; n = 2, print(j) get executed 8 times. n = 3,print(j) get executed 15 times. We calculate the sequence to be 3,8, 15..... which has the closed form as 2 n +$ n^ 2$ . Or by asymptotic run time as Θ( $n^2$ )

  fun(n) 
       m = 1 
       for i = 1 to n 
          m = m + 2 
          for j = 1 to m 
             Print(j)

Solution: The assignment m=1 takes constant time. The line m = m + 2 is run n many times, which takes Θ( n ) time. The inner print statement runs m many times, and m = 2n + 1 for each n. The total number of times the print statement runs is Can someone please help me understand why(the derivation) "and m = 2n+1 for each n."? How can we draw the connections with "the line m = m+2 is run m many times.." and "the inner print statement runs m many times" to get "and m = 2n+1 for each n."?

Answer 1

Possibly it's not for each $n$, but for each $i$.

First, try to calculate $m$ for each specific $i$:

 i | m
-------
 1 | 3
 2 | 5
 3 | 7
 4 | 9
   :
 n | 2n+1

$m$ increased by two for each loop of for i = 1 to n. Therefore $m = 2i+1$ for each $i$ in the loop.

And this is repeated from $i = 1$ to $i = n$, so in total,

$$ \sum_{i=1}^n (2i + 1) = n^2 + 2n $$