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Sequence: A list of terms or elements in order. The terms are identified using positive integers as subscripts of a: a 1 a2 a3 …an. The
domain is the set of consecutive positive integers starting with 1. The terms in a sequence can form a pattern or they can be
random.
Subscripts: Consecutive counting number subscripts, starting with 1, that are used to identify the location of the terms in the
sequence. Ex: a3 indicates the 3rd term in the sequence.
Finite Sequence: Contains a specific number of terms that can be counted. The domain is a subset of the counting or natural
numbers.
Examples:
Sequence:
Terms:
2, 5, 8, 11, 14, … 72
a1, a2, a3, a4, a5, … an
Sequence:
Terms:
5, 10, 20, 40, 80, … 640
a1, a2, a3, a4, a5, … an
Infinite Sequence: Contains an unlimited number of terms that cannot be counted. The domain is the set of all natural numbers.
Sequence:
Terms:
2, 5, 7, 12, 19, 24, 26, …
a1, a2, a3, a4, a5, a6, a7, …
Sequence:
Terms:
2, 4, 8, 16, 32, …
a1, a2 a3, a4, a5, …
Finding a Specific Term in a Sequence
Recursive Definition or Formula: In a recursive definition or formula, the first term in a sequence is given and subsequent terms are
defined by the terms before it. If an is the term we are looking for, an-1 is the term before it. To find a specific term, terms prior to it
must be found.
Ex: Find the first 3 terms in the sequence an = 3 an-1 + 4 if a1 = 5. In this example, the first term is a1 = 5. To find the 2nd and
3rd terms, n = 2, and n = 3 need to be substituted.
an-1
a1 = 5
a2 = 3(a1) + 4
So, a2 = 3(5) + 4
a2 = 19
a3 = 3(a2) + 4
So, a3 = 3(19) + 4
a3 = 61
Try these:
1.
A sequence has the recursive rule f(n) = 2f(n – 1). The first term of the sequence is 7. Write the first 3 terms of the
sequence.
2.
The recursive rule for a sequence is
3.
The first term of a sequence is 4. What is the third term of the sequence with the recursive rule f(n) = 2 f(n – 1) + 2?
. The first term is 4. What is the third term?
Sequence Notes
To write a recursive definition or formula when given several terms in the sequence, it is necessary to find an expression that is
developed by comparing the terms and finding the process required to change each term to the subsequent term.
Ex: Write a recursive definition for this sequence: -2, 4, 16, 256, …
a1 = -2. Since 4 = (-2)2, and 16 = 42, and using the last term given to us, 256 = 162, a recursive definition for this sequence
could be
an = a(n – 1)2.
Try These:
1.
Write a recursive function for the sequence: 1, -2, -5, -8.
2.
Write a function that represents the sequence: 7, 14, 21, 28, …
3.
Write a recursive rule for the sequence: 2, 4, 7, 11, 16…
Explicit Formula: If specific terms are not given, a formula, sometimes called an explicit formula, is given. It can be used by
substituting the number of the term desired into the formula for “n”. Simplify as usual.
What is the 7th term in the sequence: an = 2n – 4?
Ex 1:
Ex 2:
Since we want the 7th term, n = 7. Substitute 7 for n
and solve.
a7 = 2(7) – 4 = 10
The 7th term in the sequence is 10.
What is the 5th term in the sequence: an = 3n?
Since we want the 5th term, n = 5. Substitute 5 for n and
solve.
a 5 = 35
The 5th term in the sequence is 243.
Ex 3: What are the first 3 terms in the sequence an = n2 + 1?
3 calculations are needed: n = 1, n = 2, and n = 3
a 1 = 12 + 1 = 2
a 2 = 22 + 1 = 5
a3 = 32 + 1 = 10
The first 3 terms are 2, 5, and 10.
Try these:
4 n
.
n3
1.
Find the third term of a sequence with the explicit rule f(n) =
2.
Find the fifth term of a sequence represented by f(n) = 5n – 2.
3.
What is the fifth term of the sequence defined by f(n) = 3(n – 3)?
Explicit vs. Recursive:
A recursive rule cannot be used to find a specific term directly. To find a specific term’s value using a recursive rule, you need to
know the value of one or more of the previous terms.
Ex: Suppose you want to find the 50th term of a sequence. You would rather use an explicit rule because you only need to
substitute n = 50 into the function to find the 50th term. A recursive rule would require you to calculate 49 terms before
finding the 50th term.
Sequence Notes