Unformatted text preview: ’1 = a (b)
j =1 1 âˆ’ rn
, if r = 1,
1âˆ’r n arnâˆ’1 = na, if r = 1.
j =1 3. Prove that the determinant of a lower triangular matrix is the product of the entries on the
main diagonal. (See Exercise 5 in Section 8.3.) Use this result to then show det (In ) = 1
where In is the n Ã— n identity matrix.
4. Discuss the classic â€˜paradoxâ€™ All Horses are the Same Color problem with your classmates. 9.3 Mathematical Induction 9.3.2 579 Selected Answers
n j2 = 1. (a) Let P (n) be the sentence
j =1 n(n + 1)(2n + 1)
. For the base case, n = 1, we get
6 1
? j2 =
j =1 (1)(1 + 1)(2(1) + 1)
6 12 = 1
We now assume P (k ) is true and use it to show P (k + 1) is true. We have
k+1
? (k + 1)((k + 1) + 1)(2(k + 1) + 1)
6 j 2 + (k + 1)2 = ? (k + 1)(k + 2)(2k + 3)
6 k (k + 1)(2k + 1)
?
+(k + 1)2 =
6 (k + 1)(k + 2)(2k + 3)
6 j2 =
j =1
k
j =1 Using P (k) k (k + 1)(2k + 1) 6(k + 1)2
+
6
6
k (k + 1)(2k + 1) + 6(k + 1)2
6
(k + 1)(k (2k + 1) + 6(k + 1))
6
(k + 1) 2k 2 + 7k + 6
6
(k + 1)(k + 2)(2k + 3)
6
n j2 = By induction,
j =1 ? =
? =
? =
? =...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, RenÃ© Descartes, Euclidean geometry

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