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Unformatted text preview: 13 + 23 + 33 + · · · + (x − 1)3 =
.
(3)
2
Now add x3 to both sides of (3). Thus
13 + 23 + 33 + · · · + (x − 1)3 + x3 = (x − 1) x
2 = x2 = x2 = x2 2 + x3
2 x−1
2 +x
2 (x − 1)
+x
4
2 =
=
=
=
which contradicts (2).
QED. (x − 1) + 4x
4 x2
x2 − 2x + 1 + 4x
4
x2 2
x + 2x + 1
4
x2
2
(x + 1)
4
2
x (x + 1)
,
2 b) Proof by induction on n.
1. Base case. Prove (1) is true for n = 1. Note LHS = 13 = 1 and RHS
·2
= 122 = 1. So LHS = RHS. Done with base case.
2. Induction Hypothesis. Assume (1) is true for n = k . So we a...
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This note was uploaded on 01/31/2014 for the course MATH 187 taught by Professor Holmes during the Fall '08 term at Boise State.
 Fall '08
 HOLMES
 Math, Integers

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