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Homework Solutions for CMSC 250, Fall 2002 - Sequences and Series - Prof. Jandelyn Dawn Pl, Assignments of Discrete Structures and Graph Theory

University of MarylandDiscrete Structures and Graph Theory
Prof. Jandelyn Dawn Plane

The solutions to the problems related to sequences and series for the homework 7 of the cmsc 250 course in the fall 2002 semester. It includes the first five terms of various sequences, calculations using i and ia, and finding formulas for sums or products.

Typology: Assignments

Pre 2010

Uploaded on 02/13/2009

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CMSC250-Homework7-Fall2002
DueWednesdayOct23rdatthebeginningofyourdiscussionsection
Youmustwritethesolutionstotheproblemssingle-sidedonyourownlinedpaper,withallsheets
stapledtogether,andwithallanswerswritteninsequentialorderoryouwilllosepoints.
(1) Writethefirst5termsofeachofthefollowingsequences:
(a) ak=2k-1k≥0
(b) ak=k/5k≥1
(c) a0=0;ak=ak-1+2k≥1
(2) Foreachsub-partofquestion1,write =
3
1ii
ainitsexpandedformandcalculatetheanswer.
(3) Foreachsub-partofquestion1,write ∏
=
4
2i
i
ainitsexpandedformandcalculatetheanswer.
(4) Foreachofthefollowing,determineaformulaforthesequenceanduseittorepresentthesumorproductshown.
(a)1/4+2/9+3/16+4/25+5/36
(b)(22+1)+(33-1)+(44+1)+(55-1)+(66+1)
(c)2*5*8*11*14
(5) Rewritethefollowingsummationssotheyuseasinglesummation.Youranswerneedstoyieldthesameresultasthe
originalonesaddedtogether.Youmayalsohavestandalonetermsasneeded.
=
n
i
i
0
2+ −
=
1
1
3
n
k
k
(6) Evaluateeachofthefollowingtoasinglevalue:
(a) ∏
==
2
0
3
1
ijj
i
(b)∏
==
3
1
2
0
jij
i

Partial preview of the text

Download Homework Solutions for CMSC 250, Fall 2002 - Sequences and Series - Prof. Jandelyn Dawn Pl and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity!

CMSC 250 - Homework 7 - Fall 2002

Due Wednesday Oct 23

rd

at the beginning of your discussion section

You must write the solutions to the problems single-sided on your own lined paper, with all sheets

stapled together, and with all answers written in sequential order or you will lose points.

(1) Write the first 5 terms of each of the following sequences: (a) ak = 2k^ - 1 k ≥ 0 (b) ak = k/ 5  k ≥ 1 (c) a 0 = 0; ak = ak-1 + 2 k ≥ 1

(2) For each sub-part of question 1, write 

=

3

i 1

ai in its expanded form and calculate the answer.

(3) For each sub-part of question 1, write ∏

=

4

i 2

ai in its expanded form and calculate the answer.

(4) For each of the following, determine a formula for the sequence and use it to represent the sum or product shown.

(a) 1 / 4 + 2 / 9 + 3 / 16 + 4 / 25 + 5 / 36

(b) (2^2 +1) + (3^3 -1) + (4^4 +1) + (5^5 -1) + (6^6 +1)

(c) 2 * 5 * 8 * 11 * 14

(5) Rewrite the following summations so they use a single summation. Your answer needs to yield the same result as the original ones added together. You may also have standalone terms as needed.

=

n

i

i

0

−

=

1

1

n

k

k

(6) Evaluate each of the following to a single value:

(a)  ∏

= =

2

0

3

i j 1 j

i

(b) ∏

= =

3

1

2

j i 0 j

i