Sum of first n odd numbers in O(1) Complexity

Given the sequence of odd numbers 
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, .... 
Find the sum of first n odd numbers
Examples: 

Input : n = 2
Output : 4
Sum of first two odd numbers is 1 + 3 = 4.

Input : 5
Output : 25
Sum of first 5 odd numbers is 1 + 3 + 5 +
7 + 9 = 25

A simple solution is to iterate through all odd numbers. 

Output: 

Sum of first 20 odd numbers is 400

Time Complexity: O(n) 
Auxiliary Space : O(1)

An efficient solution is to use direct formula. To find the sum of first n odd numbers we can apply odd number theorem, it states that the sum of first n odd numbers is equal to the square of n.

(2i - 1) = n² where i varies from 1 to n

let n = 10, therefore sum of first 10 odd numbers is
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
if we apply odd number theorem:
sum of first 10 odd numbers = n * n = 10 * 10 = 100.
Below is the implementation of the above approach: 

Output: 

Sum of first 20 odd numbers is 400

Time Complexity: O(1) 
Auxiliary Space : O(1)
How does it work?
We can prove it using mathematical induction. We know it is true for n = 1 and n = 2 as sums are 1 and 4 (1 + 3) respectively.

Let it be true for n = k-1.

Sum of first k odd numbers = 
 Sum of first k-1 odd numbers + k'th odd number
= (k-1)*(k-1) + (2k - 1)
= k*k