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Explicit Formulas for Sequences 53
BIG IDEA Sequences can be thought of as functions, but they have their own notation different from other functions. If their terms are real numbers, they are real functions and can be graphed.
Recognizing and Representing Sequences
In previous mathematics courses you have seen many sequences. A sequence is an ordered list of numbers or objects. Specifically, a sequence is defined as a function whose domain is the set of all positive integers, or the set of positive integers from a to b. Each item in a sequence is called a term of the sequence. In the following Activity, you will explore a sequence.
The collections of dots below form the first five terms of a sequence of triangular arrays. The numbers of dots in each collection form a sequence of numbers.
Step 1 Complete the table to show the number of dots in each of the terms pictured. Step 2 Notice that after the first term of the sequence, each subsequent term adds a predictable and increasing number of dots to the previous term. Use this fact to complete the table for the next four terms. Step 3 This process can be continued for as long as you want. You can even think of it as going on forever. Explain why the set of ordered pairs (term number, number of dots) describes a function.
ActivityActivity
Vocabulary
sequence term of a sequence subscript index explicit formula discrete function
Mental Math
Leta = – 3 andb = 3. Evaluate. a. a^2 - b^2 b. b^2 - a^2 c. (a - b) 2 d. (b - a) 2 e. (ab) 2
Term Number
Number of Dots 1 2 3 4 5
Term Number
Number of Dots 6 7 8 9
Lesson
Explicit Formulas
for Sequences
Lesson 1-
54 Functions
Chapter 1
The sequence you explored in the Activity is the sequence of triangular numbers. This sequence defines a function whose domain is the set of all positive integers. If you call this function T , then T (1) = 1, T (2) = 3, T (3) = 6, …. A notation for sequences more common than f ( x ) notation is to put the argument in a subscript. A subscript is a label that is set lower and smaller than regular text. Using subscripts, T 1 = 1, T 2 = 3, T 3 = 6, …. The notation T 3 = 6 is read “T sub three equals six.” The subscript is often called an index because it indicates the position of the term in the sequence.
See Quiz Yourself 1 at the right.
Writing Explicit Formulas for Sequences
Many sequences can be described by a rule called an explicit formula for the n th term of the sequence. Explicit formulas are important because they can be used to calculate any term in the sequence by substituting a particular value for n.
To find an explicit formula for the n th triangular number Tn , you can use the fact that the area of a triangle is half the area of a rectangle.
Notice that each triangular array of dots can be arranged to be half of a rectangular array.
For instance, the number of dots representing the 4th triangular number is half the number of dots in a 4 by 5 rectangular array.
T 4 = 1 _ 2 · 4 · 5 = 10
You can generalize this idea to develop a formula for T (^) n.
Term Number Value of Term (number of dots) (^1) T 1 = _^12 · 1 · 2 = 1 (^2) T 2 = _^12 · 2 · 3 = 3 (^3) T 3 = _^12 · 3 · 4 = 6 (^4) T 4 = _^12 · 4 · 5 = 10 n (^) T (^) n = 1 _ 2 ·^ n^ ·^ (n^ +^ 1)
5
4
5
4
QUIZ YOURSELF 1 What are T 4 and T 5?
56 Functions
Chapter 1
Example 2
It is common for people to save money in savings accounts such as Certificates of Deposites (CDs) that yield a high interest rate paid once a year. Suppose you deposited $28,700 and expected a 4.1% interest rate to be compounded annually. Then the formula S (^) n = 28,700(1.041)n–^1 gives your total savings at any time during the year leading up to the nth anniversary. a. Compute the first five terms of the sequence. b. Compute the hundredth term of the sequence. c. What does your answer to Part b mean in the context of this problem? Solution 1 a. Define the sequence using function notation on a CAS and compute the first five values. S(1) =? S(2) =? S(3) =? S(4) =? S(5) =? b. Compute S(100) in the same way. S(100) =? c. This sequence gives the total savings at the end of the nth year. So, S(100) =?^ means that on the 100th anniversary of the account opening, there will be?^ in the account. Solution 2 a. Enter the formula into a grapher and generate a table to view the first five values. The table start value is n =?^. The increment is?^. The table end value is n =?^. b. Scroll down to see the value of S(n) when n = 100. S(100) =? c. After 100 years, there will be?^ in the account.
GUIDEDGUIDED
Rates for Certificates of Deposit
Explicit Formulas for Sequences 57
Lesson 1-
A sequence is an example of a discrete function. A discrete function is a function whose domain can be put into one-to-one correspondence with a finite or infinite set of integers, with gaps, or intervals, between successive values in the domain. The graphs of discrete functions consist of unconnected points. The gaps in the domain of a sequence are the intervals between the positive integers. The graph of gold prices on page 5 and the graph in Example 1 of this lesson are both examples of graphs of discrete functions.
Questions
COVERING THE IDEAS
- Consider the increasing sequence 1, 3, 5, 7, ... of positive odd numbers. a. 13 is the 7th?^ of the sequence. b. If this sequence is called D , what is D 11?
- Consider the equation a 11 = 22.83.
a. Which number is the subscript? b. What does the number that is not the subscript represent? c. Which term of the sequence is this? d. Rewrite the equation using function notation. e. Rewrite the equation in words.
In 3 and 4, consider the sequence T of triangular numbers in the Activity on page 53.
- Compute the 20th triangular number.
- If Tn = 15, what is the value of n?
- a. Draw a possible next term in the sequence at the right.
b. How many dots does it take to draw each of the first 5 terms? c. Determine an explicit formula for the sequence S (^) n if S (^) n = the number of dots in the n th term.
- Consider the sequence h whose first six terms are 231, 120, 91, 66, 45, 28. a. What number is the 4th term? b. How is the sentence “ h 5 = 45” read? c. h 6 =?
In 7 and 8, an explicit formula for a sequence is given. Write the first four terms of the sequence.
- a (^) n = 7.3 - 3 n
- Sn = n ____( n^ +^ 1)(2 6 n^ +^ 1)(sum of the first n squares)
Explicit Formulas for Sequences 59
Lesson 1-
a. Write the first six ordered pairs that relate the year to the number of seeds. b. Find an explicit formula for the number of seeds at the beginning of the n th year, for all n > 1.
- Some common bacterial cells, such as E. coli, can divide and double every 20 minutes. The doubling process takes place when a microbe reproduces by splitting to make 2 cells. Each of these cells then splits in half to make a total of 4 cells. Each of these 4 cells then splits to make a total of 8, and so on. Each splitting is called a generation. If a colony begins with 125 microbes, the equation P (^) n = 125(2) n -^1 gives the number of microbes in the n th generation (assuming all microbes survive). a. Calculate the first three terms of the sequence. b. Identify the independent variable and dependent variable of the function.
REVIEW
- Solve for s in the formula d = 7 s - 13. (Lesson 1-7)
- The measure θ of an exterior angle of a regular polygon is given by θ = 360 _ n , where n is the number of sides of the polygon. Solve for n in terms of θ. (Lesson 1-7)
In 17 and 18, an equation is given. (Lesson 1-6)
a. Solve the equation. b. Check your answer.
- 5 r - (2 r + 1) = 6 18. 7 t - (9 - 4 t ) = 9
- A medium pizza costs $12.50 plus $1.50 for each topping. If C is the total cost of the pizza and t is the number of toppings ordered, then C ( t ) = 12.50 + 1.50 t gives c as a function of t. (Lesson 1-4) a. Specify the domain of this function. b. Write the four smallest numbers in the range of this function. c. Graph this function.
- Fill in the Blank Let h : a → a^3 + 2. Then h : – 3 →?^. (Lesson 1-3)
EXPLORATION
- Triangular numbers have many curious properties. For example, a triangular number can never end with the digits 2, 4, 7, or 9. Find one more property of the triangular numbers.
QUIZ YOURSELF ANSWERS
- T 4 = 10; T 5 = 15
- 120; 5050
- p(n) = (^) ( _^12 )
n
E. coli cells
