What is the sum of the first one-hundred odd numbers, starting with 1?

A fascinating pattern emerges when you add consecutive odd numbers starting from one. The sum of the first two odd numbers (1 + 3) is 4, which is 2 squared. The sum of the first three (1 + 3 + 5) is 9, or 3 squared. This elegant rule holds true no matter how many odd numbers you add: the sum of the first 'n' odd numbers is always 'n' squared. Since we are looking for the sum of the first one-hundred odd numbers, the result is simply 100 squared, or 100 x 100, which equals 10,000.

Another way to visualize this is by pairing numbers. The series runs from 1 all the way to the 100th odd number, which is 199. If you pair the first and last numbers (1 + 199), they sum to 200. If you pair the second and second-to-last numbers (3 + 197), they also sum to 200. This pattern continues for the entire set. Since there are 100 numbers in total, you can make exactly 50 of these pairs. The total sum is therefore 50 pairs multiplied by 200, the value of each pair.

This clever pairing technique is a classic problem-solving strategy in mathematics. It is famously attributed to the child prodigy Carl Friedrich Gauss, who supposedly used a similar method in primary school to quickly sum all the integers from 1 to 100. Whether you use the simple squaring rule or the pairing method, both paths lead to the same impressive total.