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What is 1 + 3 + 5 + 7 + ... 99 = ?

Updated: Jul 28

There are several different ways to solve this problem, but here we would like to present to you a geometrical approach.


Let's start with the first two terms, i.e "1 + 3": 1 + 3 = 4 which is square of 2.


When we add 5 to that we get 1 + 3 + 5 = 9 and this is square of 3.


And, after adding one more term to the sum, we have 1 + 3 + 5 + 7 = 16 which is square of 4. This keeps going on like that.


So, each time we add a term we actually calculate the area of a square with one of its sides being equal to the number of terms in the sum. Below picture depicts how we construct this model squares as we keep adding terms in the sum.


This is a wonderful way of seeing the solution to this problem. Don't you agree?


We know from the relation 2n – 1 = 99 that 99 corresponds to the n=50th term in the sum. So, geometrically speaking, the sum of all the odd integer numbers up to 99 (including 99), forms a 50 by 50 square. Then, the sum simply becomes 50 x 50 = 2500.


The answer is 2500.


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