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Unformatted text preview: nduction that (1) 1 + 2 + 3 + ... + n = n(n + 1) 2 for every n ∈ N. To do so, let p(n) denote the statement (1). Then clearly p(1) is true. Suppose now that p(n) is true, so that 1 + 2 + 3 + ... + n = Then 1 + 2 + 3 + . . . + n + (n + 1) = n(n + 1) (n + 1)(n + 2) + (n + 1) = , 2 2 n(n + 1) . 2 so that p(n + 1) is true. It now follows from the Principle of induction (Weak form) that (1) holds for every n ∈ N. Example 1.2.2. We shall prove by induction that (2) 12 + 22 + 32 + . . . + n2 = n(n + 1)(2n + 1) 6 for every n ∈ N. To do so, let p(n) denote the statement (2). Then clearly p(1) is true. Suppose now that p(n) is true, so that 12 + 22 + 32 + . . . + n2 = Then 12 + 22 + 32 + . . . + n2 + (n + 1)2 = n(n + 1)(2n + 1) (n + 1)(n(2n + 1) + 6(n + 1)) + (n + 1)2 = 6 6 (n + 1)(2n2 + 7n + 6) (n + 1)(n + 2)(2n + 3) = = , 6 6 n(n + 1)(2n + 1) . 6 so that p(n + 1) is true. It now follows from the Principle of induction (Weak form) that (2) holds for every n ∈ N. Example 1.2.3. We shall p...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Math, Calculus

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