Chapter 4 Solutions
An Introduction to Mathematical Thinking:
Algebra and Number Systems
William J. Gilbert and Scott A. Vanstone, Prentice Hall, 2005
Solutions prepared by William J. Gilbert and Alejandro Morales
Exercise 4-1:
Calculate the following. 5 3
Solution:
5 3 = 5 3 · · 4 2 ...
, By definition,
n+2 (n + 2)!
= .
n 2!n!
Hence, we must solve n2 + 3n + 2 = 72 or n2 + 3n + 70 = 0. By the quadratic
formula, we find n = 7 or n = −10.
Exercise 4-8:
Write the following in sigma notation.
1 3 5 99
+ + + ··· + .
2 4 6 100
Solution:
99
1 3 5 99 X r
+ + + ··· + = .
2 4 6 100 r=1 r + 1
Exercise 4-9:
Write the following in sigma notation.
8 + 15 + 24 + 35 + · · · + (n2 − 1).
Solution:
n
X
8 + 15 + 24 + 35 + · · · + (n2 − 1) = (r2 − 1).
r=3
Exercise 4-10:
Write the following in sigma notation.
ak + a2k + a4k + a8k + a16k + · · · + a256k
Solution:
8
X r
ak + a2k + a4k + a8k + a16k + · · · + a256k = a2 k .
r=0
Exercise 4-11:
Prove, by induction, the following results for all n ∈ P.
12 + 22 + 32 + · · · + n2 = n(n+1)(2n+1)
6 .
Solution:
1(1+1)(2·1+1) 6
(i) When n = 1 the assertion is true since 6 = 6 = 1.
4.3
, (ii) Suppose that
k(k + 1)(2k + 1)
12 + 22 + 32 + · · · + k 2 = .
6
Then
12 + 22 + 32 + · · · + k 2 + (k + 1)2
k(k + 1)(2k + 1) 6(k + 1)(k + 1)
= +
6 6
(k + 1)(k(2k + 1) + 6(k + 1)
=
6
(k + 1)(k(2k + 1) + 2(2k + 1) + 2(k + 2))
=
6
(k + 1)(k + 2)(2k + 1 + 2) (k + 1)(k + 2)(2(k + 1) + 1)
= = .
6 6
Hence the assertion is true for n = k + 1.
Therefore, by the principle of mathematical induction, the assertion is true
for all n ∈ P.
Exercise 4-12:
Prove, by induction, the following results for all n ∈ P.
2
3 3 3 3 n(n + 1)
1 + 2 + 3 + ··· + n =
2
Solution: h i2
(i) If n = 1, then 13 = 1(1+1)
2 . Hence the proposition is valid for n = 1.
h i2
(ii) Assume that 13 + 23 + · · · + k 3 = k(k+1)
2 for some k ∈ P. Then
k + 4k + 4
= (k + 1)2 = (k + 1)2
4 22
2
(k + 1)(k + 2)
= .
2
Thus the proposition is valid for n = k + 1. Therefore, by the principle of
mathematical induction, the proposition is true for all n ∈ P.
Exercise 4-13:
Prove by induction the following results for all n ∈ P.
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller Examhack. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $19.49. You're not tied to anything after your purchase.
