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Examples. Identify the Sequence. Find the Next Term. Popular Problems. Identify the Sequence. Find the Next Term. Free sequence calculator - step-by-step solutions to help identify the sequence and find the nth term of arithmetic and geometric sequence types.
The geometric series you have is equivalent to evaluating 1 1 − x at x = 2; the − 1 is valid for that formula, but you are using the series outside its region of validity, which is where the trouble lies. That should beg the question of why adding up a bunch of positives results in a negative... ;) – J. M. ain't a mathematician.
The series of numbers 1, 2, 4, 8, 16 ... is an example of a geometric sequence, sometimes called a geometric progression (GP). Each term in the progression is found by multiplying the previous number by 2. Such sequences occur in many situations; the multiplying factor does not have to be 2.
Learn how to solve 1,-2,4,-8,16. Tiger Algebra's step-by-step solution shows you how to find the common ratio, sum, general form, and nth term of a geometric sequence.
- 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...
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.
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.
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