## Mathematics: The Science of Patterns

Since elementary grades, we have learned that mathematics is closely related to patterns. Given a sequence of numbers, we would know the next few terms without much effort. We know that the next three terms in the sequence 3, 7, 11, 15, are 19, 23, and 27. Just by looking at the Koch’s  snowflake shown in the first figure, we have an idea of how to draw the sixth figure. It may not be as accurate as a computer drawing, but we would surely be able to draw the necessary details. These patterns are obvious and intuitive, so it is easy to predict the next “terms.”

A little more challenging pattern is shown the second figure. Observing the colors of the circles, we can see that the sum of the first $2$ odd integers is $2^2$, the sum of the first $3$ odd integers is $3^2$, the sum of the first $4$ odd numbers is $4^2$, and so on.  From the pattern, we are quite confident that the sum of the first $1000$ odd integers is $1000^2$ without having to exhaust the 999 odd integers. » Read more