I. Recursive Descriptions A phrase like to get the next term you add 2, which tells how to obtain

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1 Mathematics 45 Describing Patterns in s Mathematics has been characterized as the science of patterns. From an early age students see patterns in mathematics, including counting by twos, threes, etc., geometric patterns, the standard computational algorithms, and much more. Students often use inductive learning by inferring patterns from examples, but unless this is enhanced with discussion, explanation, and eventually understanding, that inferred pattern may not be what was intended. (For example, an inferred pattern in some texts could be if the problem shows a rectangle and its two dimensions, then to get the answer you multiply the two numbers.) There are two standard ways to describe sequences. The first way (developmentally) emphasizes the relationships between the numbers in the sequence. It may deal with differences, quotients, or some other means of comparing consecutive terms, or it may tell how to generate succeeding terms in the sequence. ( Terms means the individual numbers in the sequence.) Here are some examples: Developmentally Simple Descriptions Possible Visualization, 2, 3, 4, 5,... Counting sequence common difference of to get the next term you add, 3, 5, 7, 9,... Odd numbers common difference of 2 to get the next term you add 2 2, 4, 6, 8, 0,... Even numbers common difference of 2 to get the next term you add 2 Note that common difference of 2 and to get the next term you add 2 are two ways of saying the same thing. I. A phrase like to get the next term you add 2, which tells how to obtain a succeeding term from the prior term (or sometimes terms) is called a recursive description (or recursive rule) for the sequence. Notice that the descriptions common difference of 2 and to get the next term you add 2 are listed for both the last two sequences. This means that both sequences have the same recursive description, so this phrase alone does not completely determine a sequence. Find two sequences to which have a common difference of 3 and to get the next term you add 3 apply: and If a recursive description for a sequence is known, the additional piece of information that is needed to completely determine the sequence is the starting, or initial, number. Once that starting point is known, each succeeding term can be generated from the previous term. Here are some examples: Initial Term Recursive Rule Terms of Add 2, 3, 5, 7, 9,... 6 Add 3 6, 9, 2, 5, 8,... 7 Add 5 7, 2, 7, 22, 27,... Now try some yourself: Mathematics 45 Describing Patterns in s Page

2 Initial Term Recursive Rule Terms of 4 Add 3 2 Add 6 3, 7,, 5, 9, 23,... All of the sequences above have a recursive description of the form to get the next term you add d. Such sequences are called arithmetic sequences, and d is called the common difference of the terms of the sequence. So one way to test whether a sequence is an arithmetic sequence is to see if the terms have a common (constant) difference. (We ll see below some recursive rules for other sequences. II Explicit Descriptions Recursive descriptions allow us to keep calculating terms as far out in the sequence as we want, but we can t jump over any terms in doing so. That is, to get the 20th term, we have to start at the beginning of the sequence and find all the terms up to the 20th. This can be very tedious. It would be nice to have a formula which would allow us to calculate any term just by knowing which term it was. Mathematically, this means we would like to have a rule or formula f (actually a function) which would take the term number n and calculate the value f(n) of term number n. Such a formula is called an explicit description (or explicit rule) for the sequence. To see how we might find an explicit rule for an arithmetic sequence, consider the sequence 5, 8,, 4, 7,.... The terms of this sequence are shown as f(n) in the second column below. In the third, fourth, and fifth columns the terms of the sequence have been reformulated using the way we got the terms in the sequence recursively. You should see a pattern in the reformulation in the last column. n f(n) How obtained Reformulation Result of reformulation 5 initial term x x x3 Notice that in the last column, f(n) has been rewritten as f(n) = 5 + (a multiple of 3) or f(n) = (initial term) + (multiple of the common difference). Notice the multiple that is used. It is always one less than the term number n. For example, in the last row of the table, where n = 5, the multiple of 3 is 4x3, where 4 = 5-. So, to get term n we should add (n - ) x 3 to the initial term. Often this is simplified to f(n) = 5 + (n - ) x 3 = 5 + 3n - 3 = 2n + 2. In general, to get the n th term of an arithmetic sequence, you start with the initial term and add the common difference (n - ) times. This often stated as follows: The explicit description of an Arithmetic is given by f(n) = (initial term) + (n - )(common difference) or f(n) = a + (n - ) x d, where a is the initial term and d is the common difference Mathematics 45 Describing Patterns in s Page 2

3 Now you try it Find an explicit description of the sequence 2, 7, 22, 27, 32, 37,.... III Geometric s Here are some other sequences that have easy recursive descriptions. 2, 4, 8, 6, 32,... To get the next term you multiply by 2 Common ratio (growth factor) of 2 4, 2, 36, 08, 324,... To get the next term you multiply by 3 Common ratio (growth factor) of 3, /2, /4, /8, /6,... To get the next term you multiply by /2 Common ratio (growth factor) of /2 The preceding three sequences all have a recursive rule which involves multiplying by a particular factor. s like this are called geometric sequences and we say that the terms of the sequence are growing exponentially. Whereas an arithmetic sequence has a common difference between the terms, a geometric sequence has a common ratio between the terms. That ratio is the amount a term must be multiplied by to obtain the next term, so this ratio is also called the growth factor of the sequence or the growth factor of the exponential growth. To find an explicit description of a geometric sequence, proceed in a manner similar to what we did with arithmetic sequences. Using the second sequence above, 4, 2, 36, 08, 324,..., write down and reformulate the terms, looking for a pattern. n f(n) How obtained Reformulation Result of reformulation 4 initial term x 3 4 x 3 4 x x 3 4 x 3 x 3 4 x x 3 4 x 3 x 3 x 3 4 x x 3 4 x 3 x 3 x 3 x 3 4 x 3 4 Compare this to what happened with an arithmetic sequence. With our geometric sequence: f(n) = 4 x (power of 3), and that power is less than the value of n, so f(n) = 4 x 3 (n-). Here 4 is the initial term, and the recursive rule is multiply by 3, where 3 is the common ratio (growth factor). Mathematics 45 Describing Patterns in s Page 3

4 In general, to get the n th term of a geometric sequence, you start with the initial term and multiply by the growth factor (n - ) times. (Compare this with what happens with arithmetic sequences!) This often stated as follows: The explicit description of an Geometric is given by f(n) = (initial term) x (common ratio) (n - ) or f(n) = a x r (n - ), where a is the initial term and r is the common ratio (growth factor) Now you try it!. 3, 6, 2, 24, 48,... Type of sequence: 2. 5, 8,, 4, 7,... Type of sequence: 3. 8, 6, 2, 2/3, 2/9,... Type of sequence: Of course, some sequences are neither arithmetic nor geometric. The main point of the above discussion is to understand the difference between an explicit description of a pattern and a recursive description of a pattern, and to be able to determine these for arithmetic and geometric sequences. Mathematics 45 Describing Patterns in s Page 4

5 IV Other s The sequence of squares and the sequence of triangular numbers, mentioned in these handouts and in the text, are neither arithmetic nor geometric sequences; they are denoted Sn and Tn, respectively. However, they still can be described both recursively and explicitly. Try to find these descriptions. Each description may need to use the term number n. Once you have done this, find the next 4 terms for the third sequence Un below, and find its recursive and explicit descriptions as well. n Recursive Description Explicit Description Sn Tn Un Mathematics 45 Describing Patterns in s Page 5