Welcome to triangular numbers! Triangular numbers are special numbers formed by adding consecutive positive integers starting from one. They get their name because when we arrange dots to represent these numbers, they naturally form triangular patterns. Let's see how this works with the first few triangular numbers.
Now let's discover the formula for triangular numbers. The nth triangular number can be calculated using the formula T sub n equals n times n plus one, divided by two. This elegant formula comes from the mathematical sum of consecutive integers from one to n. For example, the fourth triangular number is four times five divided by two, which equals ten. The fifth triangular number is five times six divided by two, which equals fifteen.
Let's look at the sequence of triangular numbers. The first ten triangular numbers are: one, three, six, ten, fifteen, twenty-one, twenty-eight, thirty-six, forty-five, and fifty-five. Notice an interesting pattern: each triangular number equals the previous triangular number plus the next integer. For example, three plus three equals six, six plus four equals ten, and so on. When we plot these points on a graph, they form a beautiful curved pattern that follows our formula.
Now let's understand why our formula works through a beautiful visual proof. We start with a triangular arrangement of dots. Next, we duplicate this triangle and flip it upside down. When we combine both triangles, they form a perfect rectangle. This rectangle has dimensions n by n plus one. Since our original triangle is exactly half of this rectangle, the triangular number equals n times n plus one, divided by two. This elegant visual proof shows why the formula works for any triangular number.
To summarize what we've learned about triangular numbers: They are formed by adding consecutive positive integers starting from one. When arranged as dots, they naturally form triangular patterns, which gives them their name. The elegant formula T sub n equals n times n plus one divided by two allows us to calculate any triangular number instantly. These fascinating numbers appear throughout mathematics and have many practical applications in combinatorics, geometry, and computer science.