Some sequences are defined by a recursion formula —that is, a formula that defines each term of the sequence in terms of one or more of the preceding terms. For example, if { a n } is defined by a 1 = 1 a n d a n = 2 a n − 1 + 1 f o r n ≥ 2 then a 2 = 2 a 1 + 1 = 2 ⋅ 1 + 1 = 3 a 3 = 2 a 2 + 1 = 2 ⋅ 3 + 1 = 7 a 4 = 2 a 3 + 1 = 2 ⋅ 7 + 1 = 15 and so on. In Problems 63–66, write the first five terms of each sequence. 66. a 1 = 1 and a n = − 1 3 a n − 1 for n ≥ 2
Some sequences are defined by a recursion formula —that is, a formula that defines each term of the sequence in terms of one or more of the preceding terms. For example, if { a n } is defined by a 1 = 1 a n d a n = 2 a n − 1 + 1 f o r n ≥ 2 then a 2 = 2 a 1 + 1 = 2 ⋅ 1 + 1 = 3 a 3 = 2 a 2 + 1 = 2 ⋅ 3 + 1 = 7 a 4 = 2 a 3 + 1 = 2 ⋅ 7 + 1 = 15 and so on. In Problems 63–66, write the first five terms of each sequence. 66. a 1 = 1 and a n = − 1 3 a n − 1 for n ≥ 2
Solution Summary: The author calculates the term for n=2 as follows.
Some sequences are defined by arecursion formula—that is, a formula that defines each term of the sequence in terms of one or more of the preceding terms. For example, if {an} is defined by
a
1
=
1
a
n
d
a
n
=
2
a
n
−
1
+
1
f
o
r
n
≥
2
then
a
2
=
2
a
1
+
1
=
2
⋅
1
+
1
=
3
a
3
=
2
a
2
+
1
=
2
⋅
3
+
1
=
7
a
4
=
2
a
3
+
1
=
2
⋅
7
+
1
=
15
and so on. In Problems 63–66, write the first five terms of each sequence.
Consider the proof below:
Proposition: If m is an even integer, then 5m +4
is an even integer.
Proof: We see that
|5m+4=10n+4
=
2(5n+2). Therefore,
5m+4 is an even integer.
**Note: you may assume the proof is valid, just poorly written.
Based upon the Section 1.3 screencast and the reading assignment, select all
writing guidelines that are missing in the proof.
Proof begins by stating assumptions
✓ Proof has an invitational tone/uses collective pronouns
Proof is written in complete sentences
Each step is justified
☐ Proof has a clear conclusion
Note: The purpose of this problem below is to use computational techniques (Excelspreadsheet, Matlab, R, Python, etc.) and code the dynamic programming ideas seen inclass. Please provide the numerical answer to the questions as well as a sample of yourwork (spreadsheet, code file, etc.).We consider an N-period binomial model with the following properties: N = 60, thecurrent stock price is S0 = 1000; on each period, the stock price increases by 0.5% whenit moves up and decreases by 0.3% when it moves down. The annual interest rate on themoney market is 5%. (Notice that this model is a CRR model, which means that thebinomial tree is recombining.)(a) Find the price at time t0 = 0 of a (European) call option with strike price K = 1040and maturity T = 1 year.(b) Find the price at time t0 = 0 of a (European) put option with strike price K = 1040and maturity T = 1 year.(c) We consider now, that you are at time t5 (i.e. after 5 periods, which represents 1month later). Assume that the stock…
4. [-/1 Points]
DETAILS
MY NOTES
SESSCALCET2 6.5.024.
Find the approximations Tη, Mn, and S, to the integral
computer algebra system.)
ASK YOUR TEACHER
PRACTICE ANOTHER
4 39
√
dx for n = 6 and 12. Then compute the corresponding errors ET, EM, and Es. (Round your answers to six decimal places. You may wish to use the sum command on a
n
Tn
Mn
Sp
6
12
n
ET
EM
Es
6
12
What observations can you make? In particular, what happens to the errors when n is doubled?
As n is doubled, ET and EM are decreased by a factor of about
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'
and Es is decreased by a factor of about
Chapter B.1 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.
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