319HW1Correction

319HW1Correction - MAT 319/320 Correction of HW1 Exercise 1...

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Unformatted text preview: MAT 319/320 Correction of HW1 Exercise 1. Page 15, #2. Proof. By induction: 1. The property is true for n = 1 , because 1 3 = bracketleftBig 1 2 . 1.2 bracketrightBig 2 2. Assume that the property is true for n , and prove that it’s true for n + 1 : 1 3 + + n 3 = bracketleftbigg 1 2 .n. ( n + 1) bracketrightbigg 2 ⇒ 1 3 + + n 3 + ( n + 1) 3 = bracketleftbigg 1 2 .n. ( n + 1) bracketrightbigg 2 + ( n + 1) 3 ⇒ 1 3 + + ( n + 1) 3 = 1 4 . ( n + 1) 2 bracketleftbig n 2 + 4( n + 1) bracketrightbig ⇒ 1 3 + + ( n + 1) 3 = 1 4 . ( n + 1) 2 [ n + 2] 2 ⇒ 1 3 + + ( n + 1) 3 = bracketleftbigg 1 2 . ( n + 1) . ( n + 2) bracketrightbigg 2 But this is exactly the property for (n+1). square Exercise 2. Page 29, #3. Proof. a) 2 x + 5 = 8 ⇒ 2 x = 3 (existence of negative elements) ⇒ x = 3/2 (existence of inverse for nonzero elements). b) add- 2 x to both sides to get x 2- 2 x = 0 . Then factor (using distributivity) to get x ( x- 2) = 0 . Conclude with theorem 2.1.3....
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319HW1Correction - MAT 319/320 Correction of HW1 Exercise 1...

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