160
CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS
11.
n
1
+
1
n
!
"
#
$
n
10
2.59374
100
2.70481
1000
2.71692
10,000
2.71815
100,000
2.71827
1,000,000
2.71828
10,000,000
2.71828
!
!
"
e
= 2.7182818…
13.
n
4
16
64
256
1024
4096
(1 +
n
)
1/
n
1.495349 1.193722 1.067399 1.021913 1.006793 1.002033
lim
n
! "
(1 +
n
)
1/
n
= 1
15.
The graphs of
y
1
=
1
+
1
n
!
"
#
$
n
,
y
2
= 2.718281828
!
e
, and
y
3
=
1
+
1
n
!
"
#
$
n
+1
for 0
#
n
#
20
are given at the right.
1
y
y
2
y
3
4
0
0
20
17.
(A)
A
=
Pe
rt
;
P
= $10,000,
r
= 5.51% = 0.0551,
t
= 10:
A
= 10,000
e
(0.0551)10
= 10,000
e
0.551
= $17,349.87
(B)
A
= $15,000,
P
= $10,000,
r
= 0.0551:
15,000 = 10,000
e
0.0551
t
e
0.0551
t
= 1.5
0.0551
t
= ln(1.5)
t
=
ln(1.5)
0.0551
!
7.36 years
19.
A
=
Pe
rt
;
A
= $20,000,
r
= 0.052,
t
= 10:
20,000 =
Pe
(0.052)10
=
Pe
0.52
P
=
20,000
e
0.52
= 20,000
e
-0.52
!
$11,890.41
21.
30,000 = 20,000
e
5
r
e
5
r
= 1.5
5
r
= ln(1.5)
r
=
ln(1.5)
5
!
0.0811 or 8.11%