Find the first five terms of each sequence.
1.
SOLUTION:
Use a1 = 16 and the recursive formula to find the next four terms.
Thefirstfivetermsare16,13,10,7,and4.
2.
SOLUTION:
Use a1 = –5 and the recursive formula to find the next four terms.
The first five terms are –5, –10, –30, –110, and –430.
Write a recursive formula for each sequence.
3.1, 6, 11, 16, ...
SOLUTION:
Subtract each term from the term that follows it.
6 – 1 = 5; 11 – 6 = 5, 16 – 11 = 5
There is a common difference of 5. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 1, and n≥ 2.
A recursive formula for the sequence 1, 6, 11, 16, …is a1 = 1, an = an – 1 + 5, n ≥ 2.
4.4, 12, 36, 108, ...
SOLUTION:
Subtract each term from the term that follows it.
12 – 4 = 8; 36 – 12 = 24, 108 – 36 = 72
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
=3; = 3 ; = 3
There is a common ratio of 3. The sequence is geometric.
Use the formula for a geometric sequence.
The first term a1 is 4, and n≥2.Arecursiveformulaforthesequence4,12,36,108,…is a1 = 4, an = 3an – 1, n≥
2.
5.BALL A ball is dropped from an initial height of 10 feet. The maximum heights the ball reaches on the first three
bounces are shown.
a. Write a recursive formula for the sequence.
b. Write an explicit formula for the sequence.
SOLUTION:
a. The sequence of heights is 10, 6, 3.6, and 2.16. Subtract each term from the term that follows it.
6 – 10 = 4; 3.6 – 6 = -2.4, 2.16 – 3.6 = -1.44
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
; ;
There is a common ratio of 0.6. The sequence is geometric.
Use the formula for a geometric sequence.
The first term a1 is 10, and n≥ 2.Arecursiveformulaforthesequence10,6,3.6,and2.16,…is a1 = 10, an =
0.6an – 1, n≥ 2.
b. Use the formula for the nth terms of a geometric sequence.
The explicit formula is an = 10(0.6)n– 1.
For each recursive formula, write an explicit formula. For each explicit formula,
write a recursive formula.
6.
SOLUTION:
The common difference is 16.
Use the formula for the nth terms of an arithmetic sequence.
The explicit formula is an = 16n – 12.
7.an = 5n + 8
SOLUTION:
Write out the first 4 terms. 13, 18, 23, 28
Subtract each term from the term that follows it.
18 – 13 = 5; 23 – 18 = 5, 28 – 23 = 5
There is a common difference of 5. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 13, and n≥ 2. A recursive formula for the explicit formula an = 5n + 8 is a1 = 13, an = an – 1 + 5,
n≥ 2.
8.an = 15(2)n – 1
SOLUTION:
Write out the first 4 terms. 15, 30, 60, 120
Subtract each term from the term that follows it.
30 – 15 = 30; 60 – 30 = 30, 120 – 60 = 60
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of 2. The sequence is geometric.
Use the formula for a geometric sequence.
The first term a1 is 15, and n≥2.Arecursiveformulafortheexplicitformula an = 15(2)n - 1 is a1 = 15, an = 2an – 1,
n≥ 2.
9.
SOLUTION:
The common ratio is 4.
Use the formula for the nth terms of a geometric sequence.
The explicit formula is an = 22(4)n– 1.
Find the first five terms of each sequence.
10.
SOLUTION:
Use a1 = 23 and the recursive formula to find the next four terms.
The first five terms are 23, 30, 37, 44, and 51.
11.
SOLUTION:
Use a1 = 48 and the recursive formula to find the next four terms.
The first five terms are 48, –16, 16, 0, and 8.
12.
SOLUTION:
Use a1 = 8 and the recursive formula to find the next four terms.
The first five terms are 8, 20, 50, 125, and 312.5.
13.
SOLUTION:
Use a1 = 12 and the recursive formula to find the next four terms.
The first five terms are 12, 15, 24, 51, and 132.
14.
SOLUTION:
Use a1 = 13 and the recursive formula to find the next four terms.
The first five terms are 13, –29, 55, –113, and 223.
15.
SOLUTION:
Use and the recursive formula to find the next four terms.
The first five terms are .
Write a recursive formula for each sequence.
16.
SOLUTION:
Subtract each term from the term that follows it.
–1 – 12 = –13; –14 – (–1)=–13, –27 – (–14) = –13
There is a common difference of –13. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 12, and n≥2.Arecursiveformulaforthesequence12,–1, –14, –27, …is a1 = 12, an = an– 1 –
13, n≥ 2.
17.27, 41, 55, 69, ...
SOLUTION:
Subtract each term from the term that follows it.
41 – 27 = 14; 55 – 41=14,69– 55 = 14
There is a common difference of 14. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 27, and n≥ 2. A recursive formula for the sequence 27, 41, 55, 69, …is a1 = 27, an = an – 1 +
14, n≥ 2.
18.2, 11, 20, 29, ...
SOLUTION:
Subtract each term from the term that follows it.
11 – 2 = 9; 20 – 11=9,29– 20 = 9
There is a common difference of 9. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 2, and n≥ 2. A recursive formula for the sequence 2, 11, 20, 29, …is a1 = 2, an = an – 1 + 9, n≥
2.
19.100, 80, 64, 51.2, ...
SOLUTION:
Subtract each term from the term that follows it.
80 – 100 = –20; 64 – 80 = –16, 51.2 – 64 = 12.8
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of 0.8. The sequence is geometric.
Use the formula for a geometric sequence.
The first term a1 is 100, and n≥ 2. A recursive formula for the sequence 100, 80, 64, 51.2, …is a1 = 100, an =
0.8an – 1, n≥ 2.
20.
SOLUTION:
Subtract each term from the term that follows it.
–60 – 40 = –100; 90 – (–60) = 30, –135 – 90 = –225
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of –1.5. The sequence is geometric.
Use the formula for a geometric sequence.
The first term a1 is 40, and n≥ 2. A recursive formula for the sequence 40, –60, 90, –135, …is a1 = 40, an = –
1.5an– 1, n≥ 2.
21.81, 27, 9, 3, ...
SOLUTION:
Subtract each term from the term that follows it.
27 – 81 = –54; 9 – 27 = –18, 3 – 9 = –6
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of .Thesequenceisgeometric.
Use the formula for a geometric sequence.
The first term a1 is 81, and n≥ 2. A recursive formula for the sequence 81, 27, 9, 3, …is a1 = 81, an = an– 1, n≥ 2.
22.CCSS MODELING A landscaper is building a brick patio. Part of the patio includes a pattern constructed from
triangles. The first four rows of the pattern are shown.
a. Write a recursive formula for the sequence.
b. Write an explicit formula for the sequence.
SOLUTION:
For each recursive formula, write an explicit formula. For each explicit formula,
write a recursive formula.
23.
SOLUTION:
Write out the first 4 terms: 3, 12, 48, 192.
Subtract each term from the term that follows it.
12 – 3 = 9; 48 – 12 = 36, 192 – 48 = 144
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of 4. The sequence is geometric.
Use the formula for a geometric sequence.
The first term a1 is 3, and n≥ 2. A recursive formula for the explicit formula an = 15(2)n – 1 is a1 = 3, an = 4an – 1, n
≥ 2.
24.
SOLUTION:
Thecommondifferenceis–12. Use the formula for the nth terms of an arithmetic sequence.
The explicit formula is an = –12n + 10.
25.
SOLUTION:
The common ratio is . Use the formula for the nth terms of a geometric sequence.
Theexplicitformulais .
26.
SOLUTION:
Write out the first 4 terms. 45, 38, 31, 24
Subtract each term from the term that follows it.
38 – 45 = –7; 31 – 38 = –7, 24 – 31 = –7
There is a common difference of –7. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 45, and n≥ 2. A recursive formula for the explicit formula an = –7n + 52 is a1 = 45, an = an – 1 –
7, n≥ 2.
27.TEXTING Barbara received a chain text that she forwarded to five of her friends.
Each of her friends forwarded the text to five more friends, and so on.
a.Findthefirstfivetermsofthesequencerepresentingthenumberofpeoplewhoreceivethetextinthenthround.
b.Writearecursiveformulaforthesequence.
c. If Barbara represents a1, find a8.
SOLUTION:
a.
Then the first 5 terms of the sequence would be 1, 5, 25, 125, 625.
b. The first term a1 is 1, and n≥ 2. A recursive formula is a1 = 1, an = 5an – 1, n≥2.
c.
Onthe8thround,78,125wouldreceivethechaintextmessage.
28.GEOMETRY Consider the pattern below. The number of blue boxes increases according to a specific pattern.
a. Write a recursive formula for the sequence of the number of blue boxes in each figure.
b. If the first box represents a1, find the number of blue boxes in a8.
SOLUTION:
a. The sequence of blue boxes is 0, 4, 8, and 12.
Subtract each term from the term that follows it.
4 – 0 = 4; 8 – 4 = 4, 12 – 8 = 4
There is a common difference of 4. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 0, and n≥ 2. A recursive formula is a1 = 0, an = an – 1 + 4, n≥ 2.
b. Use the formula for the nth terms of an arithmetic sequence.
When n=8,therewillbe28blueboxes.
29.TREE The growth of a certain type of tree slows as the tree continues to age. The heights of the tree over the past
four years are shown.
a. Write a recursive formula for the height of the tree.
b. If the pattern continues, how tall will the tree be in two more years? Round your answer to the nearest tenth of a
foot.
SOLUTION:
a. The sequence of heights is 10, 11, 12.1, and 13.31.
Subtract each term from the term that follows it.
11 – 10 = 11; 12.1 – 11 = 1.1, 13.31 – 12.1 = 1.21
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of 1.1. The sequence is geometric.
Use the formula for a geometric sequence.
The first term a1 is 10, and n≥ 2. A recursive formula for the sequence 10, 11, 12.1, 13.31, …is a1 = 10, an =
1.1an – 1, n≥ 2.
b. Use the formula for the nth terms of a geometric sequence.
Intwomoreyears,thetreewillbe16.1feettall.
30.MULTIPLE REPRESENTATIONS The Fibonacci sequence is neither arithmetic nor geometric and can be
defined by a recursive formula. The first terms are 1, 1, 2, 3, 5, 8, …
a. Logical Determine the relationship between the terms of the sequence. What are the next five terms in the
sequence?
b. Algebraic Write a formula for the nth term if a1 = 1, a2 = 1, and n≥3.
c. Algebraic Find the 15th term.
d. Analytical Explain why the Fibonacci sequence is not an arithmetic sequence.
SOLUTION:
a. Sample answer: The first two terms are 1. Starting with the third term, the two previous terms are added together
to get the next term. So, the next 5 terms after 8 is 5 + 8 or 13, 8 + 13 or 21, 13 + 21 or 34, 21 + 34 or 55, and 34 +
55 or 89.
b. The first term a1 is 1 and the second term a2 is 1, and n≥ 3. A recursive formula for Fibonacci sequence is a1 =
1, a2 = 1, an = an ÿ 2 + an ÿ 1, n≥3.
c. First,findthe13thand14thtermsbywritingout14termsofthesequence:1,1,2,3,5,8,13,21,34,55,89,144,
233,377.
Then,a13 = 233 and a14 = 377. Thus, a15 = a13 + a14 or 610.
d. Sample answer: Fibonacci sequence is not an arithmetic sequence since there is no common difference.
31.ERROR ANALYSIS Patrick and Lynda are working on a math problem that involves the sequence 2, –2, 2, –2, 2,
…. Patrick thinks that the sequence can be written as a recursive formula. Lynda believes that the sequence can be
written as an explicit formula. Is either of them correct? Explain.
SOLUTION:
Both; sample answer: The sequence can be written as the recursive formula a1 = 2, an = (–1)an – 1, n≥2.The
sequence can also be written as the explicit formula an = 2(–1)n – 1.
32.CHALLENGE Find a1 for the sequence in which a4 = 1104 and an = 4an – 1 + 16.
SOLUTION:
Find a3 first.
Find a2 next.
Find a1.
Therefore, a1 is 12.
33.CCSS ARGUMENTS Determine whether the following statement is true or false. Justify your reasoning.
There is only one recursive formula for every sequence.
SOLUTION:
False; sample answer: A recursive formula for the sequence 1, 2, 3, …can be written as a1 = 1, an = an - 1 + 1, n≥
2 or as a1 = 1, a2 = 2, an = an - 2 + 2, n≥3.
34.CHALLENGE Find a recursive formula for 4, 9, 19, 39, 79, …
SOLUTION:
Subtract each term from the term that follows it.
9 – 4 = 5; 19 – 9 = 10, 39 – 19 = 20
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is no common ratio. Therefore the sequence must be a combination of both.
From the difference above, you can see each is twice as big as the previous. So r is 2. From the ratios, if each
numerator was one less, the ratios would be 2. Thus, the common difference is 1. So, if the first term a1 is 4, and n≥
2 a recursive formula for the sequence 4, 9, 19, 39, 79, …is a1 = 4, an = 2an – 1 + 1, n≥ 2.
35.WRITING IN MATH Explain the difference between an explicit formula and a recursive formula.
SOLUTION:
Sample answer: In an explicit formula, the nth term an is given as a function of n. In a recursive formula, the nth
term an is found by performing operations to one or more of the terms that precede it.
36.Find a recursive formula for the sequence 12, 24, 36, 48, ….
A
B
C
D
SOLUTION:
Subtract each term from the term that follows it.
24 – 12 = 12; 36 – 24 = 12, 48 – 36 = 12
There is a common difference of 12. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a1 is 12, and n≥ 2. A recursive formula for the sequence 12, 24, 36, 48, …is a1 = 12, an = an – 1 +
12, n≥ 2.
Therefore,thecorrectchoiceisC.
37.GEOMETRY The area of a rectangle is 36 m4n6 square feet. The length of the rectangle is 6 m3n3 feet. What is
the width of the rectangle?
F 216m7n9 ft
G6mn3 ft
H42m7n3 ft
J30mn3 ft
SOLUTION:
Use the area formula for a rectangle to determine the measure of the width.
Therefore, the correct choice is G.
38.Find an inequality for the graph shown.
A
B
C
D
SOLUTION:
The y-intercept of the line is –4.Theslopeis2.Thegraphisshadedbelow,sotheinequalityshouldusea<or≤
symbol.Sincethelineisdashed,theinequalitydoesnotincludetheequals.Therefore,thecorrectchoiceisC.
39.Write an equation of the line that passes through (–2, –20)and(4,58).
F y = 13x+6
G y = 19x – 18
H y = 19x + 18
J y = 13x – 6
SOLUTION:
Firstfindtheslope.
Use the point-slope formula to find the equation.
Therefore, the correct choice is F.
Find the next three terms in each geometric sequence.
40.675, 225, 75, ...
SOLUTION:
Find the common ratio.
The common ratio is . Multiply by the common ratio to find next three terms. They are .
41.16, –24, 36, ...
SOLUTION:
Find the common ratio.
The common ratio is . Multiply by the common ratio to find next three terms. They are -54, 81, -121.5.
42.6, 18, 54, ...
SOLUTION:
Find the common ratio.
The common ratio is 3. Use the common ratio to find next three terms. They are 162, 324, 972.
43.512, –256, 128, ...
SOLUTION:
Find the common ratio.
The common ratio is . Use the common ratio to find next three terms. They are -64, 32, -16.
44.125, 25, 5, ...
SOLUTION:
Find the common ratio.
The common ratio is . Use the common ratio to find next three terms. They are .
45.12, 60, 300, ...
SOLUTION:
Findthecommonratio.
The common ratio is 5. Multiply by the common ratio to find next three terms. They are 1500, 7500, 37,500.
46.INVESTMENT Nicholas invested $2000 with a 5.75% interest rate compounded monthly. How much money will
Nicholas have after 5 years?
SOLUTION:
After5years,Nicholaswillhave$2664.35.
47.TOURS The Snider family and the Rollins family are traveling together on a trip to visit a candy factory. The
number of people in each family and the total cost are shown in the table below. Find the adult and children’s
admission prices.
SOLUTION:
Let a be the admissions cost for adults and c the admission cost of children.
Then 2a + 3c = 58 and 2a + c = 38.
So,ticketsforadultscost$14andticketsforchildrencost$10.
Write each equation in standard form.
48.
SOLUTION:
49.
SOLUTION:
50.
SOLUTION:
51.
SOLUTION:
52.
SOLUTION:
53.
SOLUTION:
Simplify each expression. If not possible, write simplified.
54.
SOLUTION:
55.
SOLUTION:
56.
SOLUTION:
57.
SOLUTION:
Therearenoliketerms,sotheexpressionissimplified.
58.
SOLUTION:
59.
SOLUTION:
There are no like terms, so the expression is simplified.
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7-8 Recursive Formulas
