A sequence is an ordered set of numbers where each number has a specific relationship with the number before it and the number after it. A very familiar sequence is the set of natural numbers. The set of natural numbers starts with one. Each number in the set is one more that the last number and one less than the next number.
Each member of a sequence is called a term of the sequence. The terms of the sequence of natural numbers are 1, 2, 3, …. The first term of a sequence is called an initial value. The initial value of the sequence of natural numbers is 1.
A sequence may be infinite or finite. An infinite sequence does not have a last term. The sequence of natural numbers is an infinite sequence. It has a first term, but does not have a last term. A finite sequence has a last term. The sequence 1, 3, 5, 7, 9 is a finite sequence. The last term is 9.
There are some types of sequences that are imporant in mathematics. These are:
An arithmetic sequence is a sequence that starts off with an initial value, then each term after the first is found by adding a constant to the previous term. The constant is called the common difference of the sequence.
An example of an arithmetic sequence is the sequence 1, 3, 5, 7, …. The initial value is 1. The common difference is 2. To find the common difference of an arithmetic sequence, subtract a term from the term before it. In the example above, subtract 5 from 7: 7 - 5 = 2.
To find out if a sequence is an arithmetic sequence, find the differences between all known terms and their previous term. If there is always a common difference, then the sequence is an arithmetic sequence.
An arithmetic sequence can be defined using an iterative definition. An interative definition states the first term, then how to get each term after the first. This type of definition makes use of indexed variables. An indexed variable has a letter followed by a subscript, such as a_{1}. The subscript is the order of the term, starting with 0. The first term is labeled a_{0}, the second a_{1} and so on. Take, for example, the arithmetic sequence 2, 5, 8, 11, …. The first term is 2. The common difference is 3: (5 - 2 = 3, 8 - 5 = 3, …). This is stated using a variable with an index of n. Here, n represents which term is being calculated. The definition of each term is: a_{n} = 2 + 3n, n = 0 … ∞. This says that the first term is 2 and each term is 3 more than the last.
What, then, would be the iterative definition of the sequence -6, -3, 0, 3, 6, …? The first term is -6 and the common difference is 3, so b_{n} = -6 + 3n, n = 0 … ∞. In general, any arithmetic sequence with a initial value of a and a common difference of d can be defined as x_{n} = a + dn, n = 0 … ∞. Table 2 contains a list of arithmetic sequences. Write down on a piece of paper the iterative definition of the sequence, then click on the Click to see definition box to see the correct definition.
Item | Sequence | Definition |
---|---|---|
1 | 3, 16, 29, … | Click to see definition t_{n} = 3 + 13n, n = 0 … ∞. |
2 | 1, -1, -3, -5, -7 | Click to see definition t_{n} = 1 - 2n, n = 0 … 4. This sequence is not an infinite sequence, so the iterator n is limited to 5 iterations. |
3 | 5, 10, 15, 20, … | Click to see definition q_{n} = 5 + 5n, n = 0 … ∞. Since the initial value equals the common difference, this could also be written q_{n} = 5n, n = 1 … ∞. Note that in this case the iterator starts at 1. |
4 | -2, -2.5, -3, -3.5, … | Click to see definition r_{n} = -2 - 0.5n, n = 0 … ∞. |
Table 2: Iterative defintions of arithmetic sequences. |
Arithmetic sequences can also be defined using set notation. The sequence 2, 4, 6, 8, … could be defined in set notation as { x | x_{n} = 2 + 2n, n = 0..∞ }. In general, if a is the initial value and d is the common difference, then any infinite arithmetic sequence can be defined in set notation as { x | x_{n} = a + dn, n = 0 … ∞ }
A geometric sequence starts with an initial value, then each term after the first is found by multiplying the previous term by a constant called the common ratio. Take, for example, the geometric sequence 1, 2, 4, 8, 16, …. The initial value is 1. Since each term is 2 times the term before it, the common ratio is 2.
The way to tell if a sequence is a geometric sequence is to divide each term by the term before it. If the quotient is always the same number, then the sequence is a geometric sequence. If any one of the quotients is different, then the sequence is not a geometric sequence.
A geometric sequence can be defined by defining the n^{th} term, where n is a natural number. This type of definition makes use of indexed variables. An indexed variable has a letter followed by a subscript, such as a_{1}. The subscript is the order of the term, starting with 0. The first term is labeled a_{0}, the second a_{1} and so on. Take, for example, the geometric sequence 1, 3, 9, 27, …. The first term is 1. The common ratio is 3 (3 ÷ 1 = 3, 9 ÷ 3 = 3, …). This is stated using a variable with an index of n. Here, n represents which term is being calculated. The definition of each term is: a_{n} = 1 × 3^{n}, n = 0 … ∞. This says that the first term is 1 and each subsequent term is 3 times more than the last.
What, then, would be the definition of the sequence ? The first term is 1 and the common ratio is , so . In general, any geometric sequence with a initial value of a and a common ratio of r can be defined as . Table 2 contains a list of geometric sequences. Write down on a piece of paper the definition of the sequence, then click on the Click to see definition box to see a correct definition.
Item | Sequence | Definition |
---|---|---|
1 | 2, 6, 18, 54, … | Click to see definition t_{n} = 2 × 3^{n}, n = 0 … ∞. |
2 | Click to see definition . This sequence is not an infinite sequence, so the iterator n is limited to 5 iterations. | |
3 | Click to see definition . Since the initial value equals the common ratio, this could also be written . Note that in this case the iterator starts at 1 instead of 0. | |
4 | 2, 2, 2, 2, … | Click to see definition p_{n} = 2 × 1^{n}, n = 0 … ∞. |
Table 4: Defintions of geometric sequences. |
Geometric sequences can also be defined using set notation. The sequence 5, 10, 15, 20, … could be defined in set notation as { x | x_{n} = 5 × 5^{n}, n = 0 … ∞ }. In general, if a is the initial value and r is the common ratio, then any infinite geometric sequence can be defined in set notation as { x | x_{n} = a × r^{n}, n = 0 … ∞ }
A sequence is convergent if the terms of a sequence come closer and closer to a particular number. The number that the terms come closer and closer to is called the limit of the sequence. The capital letter L is conventionally used to represent this number. The formal definition of a sequence involves very small positive numbers, called infinitesimals. The Greek letter epsilon (εspan>) is used to represent these small numbers. In the definition of a limit, the indexed variables a_{n} will be used to represent the n^{th} term of the sequence.
The limit exists if and only if, for an arbitrarily small number ε, there exists a natural number N such that for every n > N, | x_{n} - L | < ε.
This definition means that, if you pick a very small interval around L, at some point in the sequence, all the terms of the sequence are contained in the interval. If you pick an even smaller interval, at some point, all the terms of the sequence are in the smaller interval. No matter how small an interval you pick, eventually all the rest of the sequence after some point is in that interval.
If the limit L exists, we say that the sequence converges to L. If the limit L does not exist, we say that the sequence diverges.
A series is the sum of the terms of a sequence. If a sequence is defined as {1, 0.1, 0.01, …}, then the sequence for that series is 1 + 0.1 + 0.01 + … = 1.111…. A partial sum of a sequence is the sum of a finite subset of the terms of the sequence starting with the first term.
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