College Algebra 9th txtbk

568 CHAPTER 8 SEQUENCES, INDUCTION, AND PROBABILITY
DVDs Purchased Annually
Age 0 1 2 Above 2 Totals
Under 12 60 70 30 10 170
12–18 30 100 100 60 290
19–25 70 110 120 30 330
Over 25 100 50 40 20 210
Totals 260 330 290 120 1,000
Find the empirical probability that a person selected at random
(A) Is over 25 and buys exactly two DVDs annually.
(B) Is 12–18 years old and buys more than one DVD annually.
(C) Is 12–18 years old or buys more than one DVD annually.
65. QUALITY CONTROL Twelve precision parts, including two that
are substandard, are sent to an assembly plant. The plant manager
selects four at random and will return the whole shipment if one or
more of the samples are found to be substandard. What is the probability
that the shipment will be returned?
CHAPTER 8
ZZZ GROUP ACTIVITY Sequences Specified by Recursion Formulas
The recursion formula* a n 5a n1 6a n2 , together with the
initial values a 1 4, a 2 14, specifies the sequence {a n } whose
first several terms are 4, 14, 46, 146, 454, 1,394, . . . . The sequence
{a n } is neither arithmetic nor geometric. Nevertheless,
because it satisfies a simple recursion formula, it is possible to
obtain an nth-term formula for {a n } that is analogous to the nthterm
formulas for arithmetic and geometric sequences. Such an
nth-term formula is valuable because it allows us to estimate a
term of a sequence without computing all the preceding terms.
If the geometric sequence {r n } satisfies the preceding recursion
formula, then r n 5r n1 6r n2 . Dividing both sides
by r n2 leads to the quadratic equation r 2 5r 6 0, whose
solutions are r 2 and r 3. Now it is easy to check that the
geometric sequences {2 n } 2, 4, 8, 16, . . . and {3 n } 3, 9, 27,
81, . . . satisfy the recursion formula. Therefore, any sequence of
the form {u2 n v3 n }, where u and v are constants, will satisfy
the same recursion formula.
We now find u and v so that the first two terms of {u2 n v3 n }
are a 1 4, a 2 14. Letting n 1 and n 2 we see that u and v
must satisfy the following linear system:
2u 3v 4
4u 9v 14
Solving the system gives u 1, v 2. Therefore, an nth-term
formula for the original sequence is a n (1)2 n (2)3 n .
Note that the nth-term formula was obtained by solving a
quadratic equation and a system of two linear equations in two
variables.
(A) Compute (1)2 n (2)3 n for n 1, 2, . . . , 6, and compare
with the terms of {a n }.
(B) Estimate the one-hundredth term of {a n }.
(C) Show that any sequence of the form {u2 n v3 n }, where
u and v are constants, satisfies the recursion formula
a n 5a n1 6a n2 .
(D) Find an nth-term formula for the sequence {b n } that is specified
by b 1 5, b 2 55, b n 3b n1 4b n2 .
(E) Find an nth-term formula for the Fibonacci sequence.
(F) Find an nth-term formula for the sequence {c n } that is specified
by c 1 3, c 2 15, c 3 99, c n 6c n1 3c n2 10c n3 .
(Because the recursion formula involves the three terms that
precede c n , our method will involve the solution of a cubic
equation and a system of three linear equations in three
variables.)
*The program RECUR, found at the website for this book, evaluates the terms in any sequence defined by this type of recursion formula.