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A "sequence" (called a "progression" in British English) is an ordered list of numbers; the numbers in this ordered list are called the "elements" or the "terms" of the sequence. A "series" is what you get when you add up all the terms of a sequence; the addition, and also the resulting value, are called the "sum" or the "summation".
- Examples
Provides worked examples of typical introductory exercises...
- Arithmetic Series
An arithmetic series is the sum of the terms of an...
- Arithmetic & Geometric Sequences
The number added (or subtracted) at each stage of an...
- Examples
- Infinite Or Finite
- In Order
- Like A Set
- As A Formula
- Many Rules
- Notation
- Arithmetic Sequences
- Geometric Sequences
- Triangular Numbers
- Fibonacci Sequence
When the sequence goes on forever it is called an infinite sequence, otherwise it is a finite sequence
When we say the terms are "in order", we are free to define what order that is! They could go forwards, backwards ... or they could alternate ... or any type of order we want!
A Sequence is like a Set, except: 1. the terms are in order(with Sets the order does not matter) 2. the same value can appear many times (only once in Sets)
Saying "starts at 3 and jumps 2 every time" is fine, but it doesn't help us calculate the: 1. 10thterm, 2. 100thterm, or 3. nth term, where ncould be any term number we want.
But mathematics is so powerful we can find more than one Rulethat works for any sequence. So it is best to say "A Rule" rather than "The Rule" (unless we know it is the right Rule).
To make it easier to use rules, we often use this special style: So a rule for {3, 5, 7, 9, ...}can be written as an equation like this: xn= 2n+1 And to calculate the 10th term we can write: x10 = 2n+1 = 2×10+1 = 21 Can you calculate x50(the 50th term) doing this? Here is another example:
In an Arithmetic Sequence the difference between one term and the next is a constant. In other words, we just add some value each time ... on to infinity. In Generalwe can write an arithmetic sequence like this: {a, a+d, a+2d, a+3d, ... } where: 1. ais the first term, and 2. d is the difference between the terms (called the "common difference") And...
In a Geometric Sequence each term is found by multiplying the previous term by a constant. In Generalwe can write a geometric sequence like this: {a, ar, ar2, ar3, ... } where: 1. ais the first term, and 2. r is the factor between the terms (called the "common ratio") And the rule is: xn = ar(n-1) (We use "n-1" because ar0is the 1st term)
The Triangular Number Sequenceis generated from a pattern of dots which form a triangle: By adding another row of dots and counting all the dots we can find the next number of the sequence.
The next number is found by adding the two numbers before ittogether: 1. The 2 is found by adding the two numbers before it (1+1) 2. The 21 is found by adding the two numbers before it (8+13) 3. etc... Rule is xn = xn-1 + xn-2 That rule is interesting because it depends on the values of the previous two terms. The Fibonacci Sequence is numbered fro...
A sequence is a list of numbers, called terms, written in a specific order. Explicit formulas define each term of a sequence using the position of the term. See Example \(\PageIndex{1}\), Example \(\PageIndex{2}\), and Example \(\PageIndex{3}\).
A sequence is a list of numbers (or elements) that exhibits a particular pattern. Each element in the sequence is called a term. A sequence can be finite, meaning it has a specific number of terms, or infinite, meaning it continues indefinitely.
Sequences (numerical patterns) are sets of numbers that follow a particular pattern or rule to get from number to number. Each number is called a term in a pattern. Two types of sequences are arithmetic and geometric.
The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.
Oct 6, 2021 · Answer: First five terms: \ (0, 1, 3, 6, 10\); \ (a_ {100} = 4,950\) Sometimes the general term of a sequence will alternate in sign and have a variable other than \ (n\). Example \ (\PageIndex {2}\): Find the first \ (5\) terms of the sequence: \ (a_ {n}= (-1)^ {n} x^ {n+1}\). Solution:
