7.1: Recursive and Explicit Formulas
Learning objectives
 Write a recursive formula for a sequence, and use the formula to identify terms in the sequence.
 Write an explicit formula for a sequence, and use the formula to identify terms in the sequence.
 Identify a sequence as arithmetic, geometric, or neither.
Introduction
Consider a situation in which the value of a car depreciates 10% per year. If the car is originally valued at $20,000, the following year it is worth 90% of $20,000, or $18,000. After another, the value is 90% of $18,000, or $16,200. If we write the decreasing values as a list: 20,000, 18,000, 16,200 ... we have written a sequence.
A sequence is an ordered list of objects. Above our "objects" were all numbers. The simplest way to represent a sequence is by listing some of its terms. For example the sequence of odd, positive integers is shown here:
1, 3, 5, 7 ... 

In this lesson you will learn to represent a sequence in two ways. The first method requires that you know the previous term in order to find the next term in the sequence. The second method does not.
Representing a Sequence Recursively
Consider again the sequence shown above. What is the next term?
As long as you are familiar with the odd integers (i.e., you can count in 2’s) you can figure out that the next term is 9. If we want to describe this sequence in general, we can do so by stating what the first term is, and then by stating the relationship between successive terms. When we represent a sequence by describing the relationship between its successive terms, we are representing the sequence recursively.
The terms in a sequence are often denoted with a variable and a subscript. All of the terms in a given sequence are written with the same variable, and increasing subscripts. So we might list terms in a sequence as a_{1}, a_{2}, a_{3}, a_{4}, a_{5} ...
We can use this notation to represent the example above. This sequence is defined as follows:
a_{1} = 1  

a_{n} = a_{n}_{1} + 2 
At first glance this notation may seem confusing. What is important to keep in mind is that the subscript of a term represents its “place in line.” So a_{n} just means the n^{th} term in the sequence. The term a_{n}_{1} just means the term before a_{n}. In the sequence of odd numbers above, a_{1} = 1, a_{2} = 3, a_{3} = 5, a_{4} = 7, a_{5} = 9 and so on. If, for example, we wanted to find a_{10}, we would need to find the 9^{th} term in the sequence first. To find the 9^{th} term we need to find the 8^{th} term, and so on, back to a term that we know.
Example 1: For the sequence of odd numbers, list a_{6}, a_{7}, a_{8}, a_{9}, and a_{10}
Solution: Each term is two more than the previous term.
 a_{6} = a_{5} + 2 = 9 + 2 = 11
 a_{7} = a_{6} + 2 = 11 + 2 = 13
 a_{8} = a_{7} + 2 = 13 + 2 = 15
 a_{9} = a_{8} + 2 = 15 + 2 = 17
 a_{10} = a_{9} + 2 = 7 + 2 = 19
Finding terms in this sequence is relatively straightforward, as the pattern is familiar. However, this would clearly be tedious if you needed to find the 100^{th} term. We will turn to another method of defining sequences shortly. First let’s consider some more complicated sequences.
Example 2: For each sequence, find the indicated term.
a. Find the 5^{th} term for the sequence:
t_{1} = 3  

t_{n} = 2t_{n}_{1} 
b. Find the 4^{th} term for the sequence:
b_{1} = 3  

b_{n} = (b_{n}_{1})^{2} + 1 
Solution:
a. t_{5} = 48
t_{2} = 2t_{1} = 2 × 3 = 6  

t_{1} = 3  \begin{align*}\rightarrow\end{align*}  t_{3} = 2t_{2} = 2 × 6 = 12 
t_{n} = 2 × t_{n}_{1}  t_{4} = 2t_{3} = 2 × 12 = 24  
t_{5} = 2t_{4} = 2 × 24 = 48 
b. b_{4} = 677
b_{2} = (b_{1})^{2} + 1 = 2^{2} + 1 = 4 + 1 = 5  

b_{1} = 2  \begin{align*}\rightarrow\end{align*}  b_{3} = (b_{2})^{2} + 1 = 5^{2} + 1 = 25 + 1 = 26 
b_{n} = (b_{n}_{1})^{2} + 1  b_{4} = (b_{3})^{2} + 1 = 26^{2} + 1 = 676 + 1 = 677 
As you can see from just a few terms of the sequence in example 2b, the terms in a sequence can grow quickly. If we compare the growth of the terms in the sequences we have seen so far, the first example, the sequence of odd numbers, was the slowest. Its growth is linear, and it is referred to as an arithmetic sequence. Every arithmetic sequence has a common difference, or a constant difference between each term. (The common difference is analogous to the slope of a line.) The sequence of odd numbers has a common difference of 2 because for all n, a_{n}  a_{n}_{ 1} = 2. The sequence in example 2a is a geometric sequence. Every geometric sequence has a common ratio. In the sequence in example 2a, the common ratio is 2 because for all n, \begin{align*}\frac{t_{n}}{t_{n1}}=2\end{align*}. The terms of a geometric sequence follow an exponential pattern. The sequence in example 2b is neither arithmetic nor geometric, though its values follow a cubic pattern.
For any of these sequences, as noted above, determining more than a few values by hand would be time consuming. Next you will learn to define a sequence in a way that makes finding the n^{th} term faster.
Representing a sequence explicitly
When we represent a sequence with a formula that lets us find any term in the sequence without knowing any other terms, we are representing the sequence explicitly.
Given a recursive definition of an arithmetic or geometric sequence, you can always find an explicit formula, or an equation to represent the n^{th} term of the sequence. Consider for example the sequence of odd numbers we started with: 1,3,5,7,...
We can find an explicit formula for the n^{th} term of the sequence if we analyze a few terms:
 a_{1} = 1
 a_{2} = a_{1} + 2 = 1 + 2 = 3
 a_{3} = a_{2} + 2 = 1 + 2 + 2 = 5
 a_{4} = a_{3} + 2 = 1 + 2 + 2 + 2 = 7
 a_{5} = a_{4} + 2 = 1 + 2 + 2 + 2 + 2 = 9
 a_{6} = a_{6} + 2 = 1 + 2 + 2 + 2 + 2 + 2 = 11
Note that every term is made up of a 1, and a set of 2’s. How many 2’s are in each term?
a_{1}  = 1  

a_{2}  = 1 + 2 = 3  
a_{3}  = 1 + 2 × 2 = 5  
a_{4}  = 1 + 3 × 2 = 7  
a_{5}  = 1 + 4 × 2 = 9  
a_{6}  = 1 + 5 × 2 = 11 
The n^{th} term has (n  1)2 ’s. For example, a_{99} = 1 + 98 × 2 = 197 . We can therefore represent the sequence as a_{n} = 1 + 2(n  1). We can simplify this expression:
a_{n}  =1+2(n1)  

a_{n}  =1 + 2n  2  
a_{n}  =2n1 
In general, we can represent an arithmetic sequence in this way, as long as we know the first term and the common difference, d. Notice that in the previous example, the first term was 1, and the common difference, d, was 2. The n^{th} term is therefore the first term, plus d(n1):
a_{n}  =a_{1}+d(n1) 

You can use this general equation to find an explicit formula for any term in an arithmetic sequence.
Example 3: Find an explicit formula for the nth term of the sequence 3,7,11,15... and use the equation to find the 50^{th} term in the sequence.
Solution: a_{n}=4n1 , and a_{50}=199
The first term of the sequence is 3, and the common difference is 4.
a_{n}  =a_{1}+d(n1)  

a_{n}  =3+4(n1)  
a_{n}  =3+4n4  
a_{n}  =4n1  
a_{50}  =4(50)1=2001=199 
We can also find an explicit formula for a geometric sequence. Consider again the sequence in example 2a:
t_{2} = 2t_{1} = 2 × 3 = 6  

t_{1} = 3  \begin{align*}\rightarrow\end{align*}  t_{3} = 2t_{2} = 2 × 6 = 12 
t_{n} = 2 × t_{n}_{1}  t_{4} = 2t_{3} = 2 × 12 = 24  
t_{5} = 2t_{4} = 2 × 24 = 48 
Notice that every term is the first term, multiplied by a power of 2. This is because 2 is the common ratio for the sequence.
t_{1}  =3  

t_{2}  =2 × 3 = 6  
t_{3}  =2 × 2 × 6 = 2^{2} × 6 = 12  
t_{4}  =2 × 2 × 2 × 6=2^{3} × 6 = 24  
t_{5}  =2 × 2 × 2 × 26 = 2^{4} × 6 = 48 
The power of 2 in the n^{th} term is (n1). Therefore the n^{th} term in this sequence can be defined as: t_{n}=3(2^{n}^{1}). In general, we can define the n^{th} term of a geometric sequence in terms of its first term and its common ratio, r:
t_{n}  =t_{1}(r^{n}^{1}) 

You can use this general equation to find an explicit formula for any term in a geometric sequence.
Example 4: Find an explicit formula for the n^{th} term of the sequence 5,15,45,135... and use the equation to find the 10^{th} term in the sequence.
Solution: a_{n}=5 × 3_{1},and a_{10} = 98,415
 The first term in the sequence is 5, and r = 3.
a_{n}  =a_{1} × r^{n}^{1}  

a_{n}  =5 × 3^{n1}  
a_{10}  =5 × 3^{101}  
a_{10}  =5 × 3^{9} = 5 × 19,683 = 98,415 
Again, it is always possible to write an explicit formula for terms of an arithmetic or geometric sequence. However, you can also write an explicit formula for other sequences, as long as you can identify a pattern. To do this, you must remember that a sequence is a function, which means there is a relationship between the input and the output. That is, you must identify a pattern between the term and its index, or the term’s “place” in the sequence.
Example 5: Write an explicit formula for the nth term of the sequence 1,(1/2),(1/3),(1/4)...
Solution: a_{n}=(1/n)
Initially you may see a pattern in the fractions, but you may also wonder about the first term. If you write 1 as (1/1), then it should become clear that the n^{th} term is (1/n).
Lesson Summary
In this lesson you have learned to represent a sequence in three ways. First, you can represent a sequence just by listing its terms. Second, you can represent a sequence with a recursive formula. Given a list of terms, writing a formula requires stating the first term and the relationship between successive terms. Finally, you can write a sequence using an explicit formula. Doing this requires identifying a pattern between n and the n^{th} term.
In this lesson we have looked at several kinds of sequences: arithmetic, geometric, and sequences that do not follow either pattern. In the remainder of the chapter you will again see these three general categories of sequences.
Points to Consider
 What is the difference between a recursive and an explicit representation of a sequence?
 How many terms do sequences have?
 What happens if we add up the terms in a sequence?
Review Questions
 Find the value of a_{6} , given the sequence defined as:

a_{1} =4 a_{n} =5a_{n}_{1}  Find the value of a_{5}, given the sequence defined as:

a_{1} =32 a_{n} =(1/2)a_{n}_{1}  Find the value of a_{n}_{1} , given the sequence defined as:

a_{1} =1 a_{n} =3a_{n}_{1}n  Consider the sequence: 2,9,16... Write an explicit formula for the sequence, and use the formula to find the value of the 20^{th} term.
 Consider the sequence: 5,10,20... Write an explicit formula for the sequence, and use the formula to find the value of the 9_{th} term.
 Consider the sequence (1/2)(1/4)(1/8) Write an explicit formula for the sequence, and use the formula to find the value of the 7_{th} term.
 Identify all sequences in the previous six problems that are geometric. What is the common ratio in each sequence?
 Consider the situation in the introduction: a car that is originally valued at $20,000 depreciates by 10% per year. What kind of sequence is this? What is the value of the car after 10 years?
 In a particular arithmetic sequence, the second term is 4 and the fifth term is 13. Write an explicit formula for this sequence.
 The membership of an online dating service increases at an average rate of 8% per year. In the first year, there are 500 members. a. How many members are there in the second year? b. How many members are there in the eighteenth year?
Review Answers
 The 6^{th} term is 12,500
 The 5^{th} term is 2
 The 7^{th} term is 178

a_{n} =7n5 a_{20} =135 
a_{n} =5 × 2^{n}^{1} a_{9} =1280 
\begin{align*}a_n\end{align*} \begin{align*}= \frac {1}{2^n}\end{align*} \begin{align*}a_{7}\end{align*} \begin{align*}= \frac {1}{128}\end{align*}  The sequence in question 1 has r = 5. The sequence is question 2 has r= 1/2. The sequence in question 5 has r = 2. The sequence in question 6 has r= 1/2.
 The sequence is a geometric sequence. The value of the car after 10 years is approximately $7748.
 a_{n}=3n2
 a. 540 members b. Approximately 1,998 members
Vocabulary
 Explicit formula
 An explicit formula for a sequence allows you to find the value of any term in the sequence.
 Natural numbers
 The natural numbers are a subset of the integers: {1,2,3,4,5....}
 Recursive formula
 A recursive formula for a sequence allows you to find the value of the n^{th} term in the sequence if you know the value of the(n1)^{th} term in the sequence.
 Sequence
 A sequence is an ordered list of objects.