Triangular Number Equation: aT(b) + b^2T(a-1) = bT(a) + a^2T(b-1)

Thread starter ramsey2879
Start date Dec 29, 2009
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Numbers

In summary, triangular numbers are a sequence of numbers that form an equilateral triangle and can be found by summing all the natural numbers from 1 to n. The formula for finding the nth triangular number is n(n+1)/2 and they have been studied for centuries with various applications in mathematics and real-world phenomena. Some interesting properties include their ability to express any positive integer as the sum of two or more consecutive triangular numbers and the fact that the difference between consecutive triangular numbers is always a perfect square. They are also used in various fields of science, such as physics, computer science, and the arts.

Dec 29, 2009

ramsey2879

Find all a,b such that a*T(b) + b^2*T(a-1) = b*T(a) + a^2*T(b-1)

example solution a = 3, b = 7

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Dec 29, 2009

CRGreathouse

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I think this holds for any (a, b).

Dec 30, 2009

hamster143

Correct, both sides are always equal to T(a*b).

What are triangular numbers?

Triangular numbers are a sequence of numbers where the pattern forms an equilateral triangle. Each number in the sequence is the sum of all the natural numbers from 1 to n, where n is the position of the number in the sequence.

How do you find the nth triangular number?

The formula for finding the nth triangular number is n(n+1)/2. For example, the 5th triangular number would be (5x6)/2 = 15.

What is the significance of triangular numbers?

Triangular numbers have been studied for centuries and have many applications in mathematics, such as in Pascal's triangle and in geometric series. They also have connections to real-world phenomena, such as the number of objects in a triangular arrangement.

What are some interesting properties of triangular numbers?

One interesting property is that every positive integer can be expressed as the sum of two or more consecutive triangular numbers. Another is that the difference between consecutive triangular numbers is always a perfect square.

How are triangular numbers used in other fields of science?

Triangular numbers have applications in many areas of science, such as in physics for calculating the number of atoms in a triangular lattice and in computer science for optimizing algorithms. They also have connections to music, art, and architecture.