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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
If we now use the inductive hypothesis from part b, we obtain
1 + 2 + 2
2
+ 2
3
+
::::
+ 2
n1
+ 2
n
= ð2
n
1Þ+ 2
n
= 2 × 2
n
1 = 2
n+1
1:
So, the statement is true for n + 1. Thus, by the Principle of Mathematical Induction the statement
is true for all k 1.
25. Let us prove this statement by induction.
a. We will begin by proving that the statement is true for k = 1. Indeed when k = 1, 9
1
1 = 8, and 8
is divisible by 8.
b. Assume that the statement is true for a generic number n > 1. So, 9
n
1 = 8q for some integer
number q. This can also be rewritten as 9
n
= 1 + 8q.
c. Prove that the statement is true for n +1. B y performing some algebra and using the inductive ...
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Publisher Resources

ISBN: 9780123822178