Chapter 7- counting techniques
... formed from a collection of n objects where the first object appears k1 times, the second object appears k2 times, and so on, is: n! ...
... formed from a collection of n objects where the first object appears k1 times, the second object appears k2 times, and so on, is: n! ...
Sets, Functions and Euclidean Space
... if A ⊆ B and B ⊆ C, then A ⊆ C. The empty set, ∅, is a set with no elements at all and it is a subset of every set. The collection of all subsets of a set A is also a set, called the power set of A and denoted by P(A). Thus, B ∈ P(A) ⇐⇒ B ⊆ A. The order of the elements in a set specification such as ...
... if A ⊆ B and B ⊆ C, then A ⊆ C. The empty set, ∅, is a set with no elements at all and it is a subset of every set. The collection of all subsets of a set A is also a set, called the power set of A and denoted by P(A). Thus, B ∈ P(A) ⇐⇒ B ⊆ A. The order of the elements in a set specification such as ...
Section 9.6 Sequences
... • Def: A geometric sequence starts with any number as the 1st entry, and then each subsequent entry is obtained by multiplying or dividing by some fixed number. This fixed number, when using multiplication, is called the ratio. • Ex’s: ...
... • Def: A geometric sequence starts with any number as the 1st entry, and then each subsequent entry is obtained by multiplying or dividing by some fixed number. This fixed number, when using multiplication, is called the ratio. • Ex’s: ...
Arithmetic Sequences
... pair of adjacent terms are not all the same, the sequence is not arithmetic. An arithmetic sequence is determined completely by the first term a, and the common difference d. Thus, if we know the first two terms of an arithmetic sequence, then we can find the equation for the nth term. Finding the T ...
... pair of adjacent terms are not all the same, the sequence is not arithmetic. An arithmetic sequence is determined completely by the first term a, and the common difference d. Thus, if we know the first two terms of an arithmetic sequence, then we can find the equation for the nth term. Finding the T ...
Searching for the Pattern
... This pattern consists of four shapes that repeat in the same order. The repeating pattern is: ...
... This pattern consists of four shapes that repeat in the same order. The repeating pattern is: ...
Sequence
In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them into computer memory; infinite sequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.