Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12 Review Exercises
c. If the height is less than 6 inches or 0.5 feet,
then:
n
⎛3
⎞
0.5 ≥ 20⎜ ⎟
⎝ 4 ⎠
n
⎛3
⎞
0.025 ≥ ⎜ ⎟
⎝4
⎠
⎛3
⎞
log ( 0.025)
≥ nlog ⎜ ⎟
⎝ 4 ⎠
log ( 0.025)
n ≥
≈12.82
⎛3
⎞
log ⎜ ⎟
⎝ 4 ⎠
The height is less than 6 inches after the
13th strike.
d. Since this is a geometric sequence with
r < 1 , the distance is the sum of the two
infinite geometric series - the distances
going down plus the distances going up.
Distance going down:
20 20
S down = = = 80 feet.
⎛ 3⎞ ⎛1⎞
⎜1−
⎟ ⎜ ⎟
⎝ 4 ⎠ ⎝ 4 ⎠
Distance going up:
15 15
S up = = = 60 feet.
⎛ 3⎞ ⎛1⎞
⎜1−
⎟ ⎜ ⎟
⎝ 4 ⎠ ⎝ 4 ⎠
The total distance traveled is 140 feet.
68. a. Since the interest rate is 6.75% per annum
compounded monthly, this is equivalent to a
rate of (6.75/12)% each month. Defining a
recursive sequence, we have:
A0
= 190,000
⎛ 0.0675 ⎞
An
= ⎜1+ ⎟An
− 1 −1232.34
⎝ 12 ⎠
⎛ 0.0675 ⎞
⎜1+ ⎟ 190,000 −1232.34
⎝ 12 ⎠
= $189,836.41
b. ( )
c. Enter the recursive formula in Y= and create
the table:
d. Scroll through the table:
After 252 months, the balance is below
$100,000. The balance is about $99,540.
e. Scroll through the table:
The loan will be paid off after 360 months (or
30 years). They will make 359 payments of
$1232.34 plus a last payment of $1221.27 plus
interest. The total amount paid is:
⎛ 0.0675 ⎞
359(1232.34) + 1221.27⎜1+
⎟
⎝ 12 ⎠
≈ $443,638.20
f. The total interest expense is the difference
of the total of the payments and the original
loan: 443,638.20 − 190,000 = $253,638.20 .
g. (a) Since the interest rate is 6.75% per annum
compounded monthly, this is equivalent to
a rate of (6.75/12)% each month.
Defining a recursive sequence, we have:
A0
= 190,000
⎛ 0.0675 ⎞
An
= ⎜1+ ⎟An
− 1 −1332.34
⎝ 12 ⎠
⎛ 0.0675 ⎞
⎜1+ ⎟ 190,000 −1332.34
⎝ 12 ⎠
= $189,736.41
(b) ( )
(c) Enter the recursive formula in Y= and
create the table:
1279
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