How can I answer so quickly and whats the correct answer?

Question

I remember during the school days , the teacher asked the class "can you tell me the sum of the first 50 odd numbers ?".
I ran quickly to the teacher and told her "the answer is 2500". The teacher replied "lucky guess".
She then asked me "can you tell me the sum of first 75 odd numbers ?".
I wait for approx 10 seconds and replied with the correct answer.

How can I answer so quickly and whats the correct answer ?

Answer 1

odd numbers add up to a perfect square
the sum of the first 10 odd number is 100 or 10^2
similarly of the first 50 odd number is 50^2 or 2500
so you just need to calculate the squares of a number, so just square 75
which is 5625

proof using arithmetic progression if you are interested -
a = 1
d = 2
n = number of odd numbers

the formula for the sum of term = n/2 * { 2*a + (n-1)*d }
subtituting the values
sum = n/2 * { 2*1 + (n-1)*2 }
so sum is = n/2*{2*n}
sum = n^2

Answer 2

The first 50 odd numbers make an aritmetic progression
( 1, 3, 5..............99)
The sum of an arithmetic progression is calculated, as follows:
S = n/2 (a + Tn) where n= quantity of numbers, a = first number, Tn = last number
Therefore, S = 50/2 ( 1 + 99) = 2500
If we want to know the sum of the first 75 odd numbers quickly, then
S = 75/ 2 ( 1 + 149) = 5625
The answer is 5625