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**Unformatted text preview: **+ (j − 1)d) =
j =1 n
(2a + (n − 1)d).
2 2. For a complex number z , (z )n = z n for n ≥ 1.
3. 3n > 100n for n > 5.
4. Let A be an n × n matrix and let A be the matrix obtained by replacing a row R of A with
cR for some real number c. Use the deﬁnition of determinant to show det(A ) = c det(A).
Solution.
1. We set P (n) to be the equation we are asked to prove. For n = 1, we compare both sides of
the equation given in P (n)
1
? (a + (j − 1)d) =
j =1 1
(2a + (1 − 1)d)
2 1
(2a)
2
a=a
? a + (1 − 1)d = 2
3 Falling dominoes is the most widely used metaphor in the mainstream College Algebra books.
This is how Carl climbed the stairs in the Cologne Cathedral. Well, that, and encouragement from Kai. 9.3 Mathematical Induction 575 This shows the base case P (1) is true. Next we assume P (k ) is true, that is, we assume
k (a + (j − 1)d) =
j =1 k
(2a + (k − 1)d)
2 and attempt to use this to show P (k + 1) is true. Namely, we must show
k+1 (a + (j − 1)d) =
j =1 k+1
(2a + (k + 1 − 1)d)
2 To see h...

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