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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
If you take the "2" on the right-hand side of the "equals"...
- 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...
- Which list of numbers makes a sequence? 6, 3, 10, 14, 15, _ _ _ _ _ _ 4,7, 10, 13, _ _ _ _ _ _ Solution. The first list of numbers does not make a sequence because the numbers lack proper order or pattern.
- Find the missing terms in the following sequence: 8, _, 16, _, 24, 28, 32. Solution. Three consecutive numbers, 24, 28, and 32, are examined to find this sequence pattern, and the rule obtained.
- What is the value of n in the following number sequence? 12, 20, n, 36, 44, Solution. Identify the pattern of the sequence by finding the difference between two consecutive terms.
- By taking an example of the number sequence: 3, 8, 13, 18, 23, 28…… The common difference is found as 8 – 3 = 5; The first term is 3. For instance, to find the 5 term using the arithmetic formula; Substitute the values of the first term as 3, common difference as 5, and the n=5.
Fibonacci sequence: a n+2 = a n+1 + a n. The first two terms are 0 and 1. Square number sequence: a n = n 2. Cube number sequence: a n = n 3. Triangular number sequence: a n = ∑ k=1 n n. This can be further evaluated using the sum of natural numbers formula.
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. An arithmetic sequence is a number pattern where the rule is addition or subtraction. To create the rule, look for the common ...
Example: Find the missing terms in the following sequence: 8, ______, 16, ______, 24, 28, 32. Solution: To find the pattern, look closely at 24, 28 and 32. Each term in the number sequence is formed by adding 4 to the preceding number. So, the missing terms are 8 + 4 = 12 and 16 + 4 = 20. Check that the pattern is correct for the whole sequence ...
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A sequence is a function whose domain is a subset of the counting numbers. The sequence established by the number of hits on the website is. \ (\ { 2,4,8,16,32,…\}\) The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term.
