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134 The Nuts and Bolts of Proof, Third Edition
always positive, but/'(x) might be negative, even when/(x) is positive.
Consider the function f{x) = x^ -h 1. Its derivative is f'{x) = 2x,
which is negative for x < 0.
MATHEMATICAL INDUCTION
1. (Note that the sum on the left-hand side of the equation involves
exactly k numbers.)
a. Is the statement true for k—Yl Yes, because 1 = 2^ - 1.
b. Let us assume that the equaUty is true for k = n. So
1 + 2 + 2^ + 2^ + • - + 2"-^, = 2" - 1.
n numbers
c. Let us check if the equaUty holds for n+ 1:
1 + 2 + 2^ + 2^ + - • + 2^-^ + 2", = 2"+^ - L
(n+1) numbers
Using the associative property of addition we can write:
1 + 2 + 2^ + 2^ + • • • + 2"-^ 4- 2" = (1 + 2 + 2^ 4- 2^ + ... 4- 2""^) + 2".
If we now use the inductive hypothesis from part b, we obtain
1 + 2 + 2^ -f 2^ + ... + 2"-^ + 2" = (2" - 1) + 2"
= 2 X 2" - 1 = 2"+i - L
So the statement is true for n+L Thus, by the principle of
mathematical induction, the statement is true for all /c > L
2. Let us prove this statement by induction.
a. We will begin by proving that the statement is true for fc = L Indeed,
when /c= 1, 9^ - 1 = 8, and 8 is divisible by 8.
b. Assume that the statement is true for fc = n. So, 9" - 1 = 8^ for some
integer number q.
c. Prove that the statement is true for n+L By performing some
algebra, we obtain
9«+i _ 1 ^ 9«+i _ 1 = 9(9") _ 1
= 9(9"-1)+ 9-L