Review Flashcard - The Number Devil Nights 7, 8, & 9

Front

What are triangle numbers?

Back

Triangle numbers are a sequence of numbers that can be arranged in the shape of an equilateral triangle. The nth triangle number is the sum of the first n natural numbers.

Front

How can triangle numbers be visually represented?

Back

Triangle numbers can be visually represented by stacking dots or objects in a pyramid shape, where each row contains one more dot than the previous row.

Front

What is the formula for calculating the nth triangle number?

Back

The formula for the nth triangle number is T(n) = n(n + 1)/2.

Front

What are irrational numbers?

Back

Irrational numbers are numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions.

Front

Why are irrational numbers referred to as 'unpleasant numbers'?

Back

Irrational numbers are referred to as 'unpleasant numbers' because they are difficult to predict and understand due to their non-repeating and non-terminating nature.

Front

What is the Golden Ratio?

Back

The Golden Ratio, often denoted by the Greek letter phi (φ), is an irrational number approximately equal to 1.6180339887, which has many aesthetic and mathematical properties.

Front

How does the Golden Ratio appear in nature?

Back

The Golden Ratio appears in nature in various forms, such as the arrangement of leaves, the branching of trees, and the pattern of petals in flowers.

Front

What is the significance of patterns in understanding triangle numbers?

Back

Recognizing patterns in triangle numbers helps in understanding their formation and in predicting subsequent triangle numbers.

Front

Can you give an example of the first few triangle numbers?

Back

The first few triangle numbers are 1, 3, 6, 10, 15, and 21.

Front

How do you derive the nth triangle number from the previous one?

Back

To derive the nth triangle number from the (n-1)th triangle number, you add n to the (n-1)th triangle number: T(n) = T(n-1) + n.