What are the characteristics of Fibonacci sequences?
Fibonacci sequences are characterised by each number being the sum of the two preceding ones, starting from 0 and 1. The sequence follows the pattern 0, 1, 1, 2, 3, 5, 8, 13, 21, and so forth, prominently featuring in various natural and mathematical contexts.
What are examples of arithmetic and geometric sequences?
Examples of arithmetic sequences include 2, 5, 8, 11, 14 (adding 3 each time), and 10, 7, 4, 1, -2 (subtracting 3 each time). Examples of geometric sequences are 2, 6, 18, 54 (multiplying by 3 each time), and 81, 27, 9, 3 (dividing by 3 each time).
What is the definition of a harmonic sequence and its properties?
A harmonic sequence is a sequence of numbers formed by taking the reciprocals of an arithmetic sequence. Its properties include the fact that the difference between consecutive terms decreases as the sequence progresses, and it diverges, meaning it does not converge to a finite limit as it extends indefinitely.
What are the differences between Pascal's triangle and the triangular number sequence?
Pascal's triangle is a triangular array of the binomial coefficients, while the triangular number sequence consists of numbers representing the total number of dots that can form an equilateral triangle. Pascal's triangle has rows of numbers with various properties and relationships, whereas the triangular number sequence is a linear series of sums.
What are the key features of an arithmetic progression and how is it calculated?
An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. It is calculated using the formula \(T_n = a + (n-1)d\), where \(T_n\) is the n-th term, \(a\) is the first term, and \(d\) is the common difference.