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Adv Alegbra HW Solutions 13

Adv Alegbra HW Solutions 13 - n th roots gives a 1 a 2...

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13 (ii) Of all the triangles in the plane having perimeter p , prove that the equilateral triangle has the largest area. Solution. Use Heron’s formula : if a triangle T has area A and sides of lengths a , b , c , then A 2 = s ( s a )( s b )( s c ), where s = 1 2 ( a + b + c ) . The inequality of the means gives ( s a ) + ( s b ) + ( s c ) 3 3 ( s a )( s b )( s c ) = A 2 s , with equality holding if and only if s a = s b = s c . Thus, equality holds if and only if a = b = c , which is to say T is equilateral. 1.27 Prove that if a 1 , a 2 , . . . , a n are positive numbers, then ( a 1 + a 2 + · · · + a n )( 1 / a 1 + 1 / a 2 + · · · + 1 / a n ) n 2 . Solution. By the inequality of the means, [ ( a 1 + a 2 + · · · + a n )/ n ] n a 1 · · · a n and [ ( 1 / a 1 + 1 / a 2 + · · · + 1 / a n )/ n ] n 1 / a 1 · · · 1 / a n . Now use the general fact that if p q > 0 and p q > 0, then pp qq to obtain ( a 1 + a 2 + · · · + a n )/ n ] n [ ( 1 / a 1 + 1 / a 2 + · · · + 1 / a n )/ n ] n ( a 1 · · · a n )( 1 / a 1 · · · 1 / a n ) . But the right side is a 1 · · · a n ( 1 / a 1 ) · · · ( 1 / a n ) = 1, so that ( a 1 + a 2 + · · · + a n )/ n ] n [ ( 1 / a 1 + 1 / a 2 + · · · + 1 / a n )/
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Unformatted text preview: n th roots gives ( a 1 + a 2 +···+ a n )( 1 / a 1 + 1 / a 2 +···+ 1 / a n ) ≥ n 2 . 1.28 True or false with reasons. (i) For all integers r with 0 < r < 7, the binomial coef f cient ( 7 r ) is a multiple of 7. Solution. True. (ii) For any positive integer n and any r with 0 < r < n , the binomial coef f cient ( n r ) is a multiple of n . Solution. False. (iii) Let D be a collection of 10 different dogs, and let C be a collection of 10 different cats. There are the same number of quartets of dogs as there are sextets of cats. Solution. True. (iv) If q is a rational number, then e 2 π iq is a root of unity. Solution. True....
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