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- Arithmetic sequence: a n = a + (n - 1) d, where a = the first term and d = common difference. Geometric sequence: a n = ar n-1, where a = the first term and r = common ratio. Fibonacci sequence: a n+2 = a n+1 + a n.
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Number sequences are sets of numbers that follow a pattern or a rule. If the rule is to add or subtract a number each time, it is called an arithmetic sequence. If the rule is to multiply or ...
- Test
GCSE; AQA; Sequences - AQA Test questions. Sequences can be...
- Algebraic Fractions
To simplify this, look for the highest common factor close...
- Quadratic Graphs
Learn about and revise quadratic, cubic, reciprocal and...
- Expressions
An equation states that two expressions are equal in value,...
- Sequences
Revise. Test. Sequences. Number sequences are sets of...
- Test
Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci Sequences, quadratic sequences, and simple geometric progressions. Generate terms of a sequence from either a term-to-term or a position-to-term rule.
- 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...
Learn how to find explicit formulas for arithmetic sequences. For example, find an explicit formula for 3, 5, 7,...
- An explicit formula directly calculates the term in the sequence that you want. A recursive formula calculates each term based upon the value of th...
- The reason we use a(n)= a+b( n-1 ), is because it is more logical in algebra
- Your shortcut is derived from the explicit formula for the arithmetic sequence like 5 + 2(n – 1) = a(n). Plug your numbers into the formula where x...
- the common difference is constant so we know the sequence will either get bigger or smaller, the tenth term is bigger than the first term so we kno...
- It'll probably help. Most of this kind of stuff is less for real world applications but for creating foundations of understanding for later skills...
- m + Bn and A + B(n-1) are both equivalent explicit formulas for arithmetic sequences. A + B(n-1) is the standard form because it gives us two usefu...
- 3 + 2(𝑛 − 1) = [distribute the 2] = 3 + 2𝑛 − 2 = [add 2 and subtract 2] = 3 + 2 + 2𝑛 − 2 − 2 = 5 + 2𝑛 − 4 = [factor out 2] = 5 + 2(𝑛 − 2)
- warning: long answer this isn't an arithmetic ("linear") sequence because the differences between the numbers are different (5-2=3, 10-5=5, 17-10=7...
Revise. Test. Sequences. Number sequences are sets of numbers that follow a pattern or a rule. If the rule is to add or subtract a number each time, it is called an arithmetic sequence....
A sequence is defined by the recurrence relation \({U_{n + 1}} = 3{U_n}\) and has \({U_0} = 1\). a) Find the first five terms of the sequence. b) Determine the formula for \({u_n}\).
To find a missing number in a Sequence, first we must have a Rule. Sequence. A Sequence is a set of things (usually numbers) that are in order. Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for a more in-depth discussion. Finding Missing Numbers
