M413test1s

# M413test1s - M413 Test 1 Solutions 1 Use induction to prove...

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M413 Test 1 Solutions 1 . Use induction to prove that 5 + 11 + 17 + ··· +(6 n - 1)=3 n 2 +2 n for all n N . First, this is true for n = 1: 6(1) - 1 = 3(1 2 ) + 2(1). Now assume the statement is true for n . So we assume that 5+11+17+ ··· +(6 n - 1) = 3 n 2 +2 n. Our goal is to show the statement is true for n + 1. That would be the statement 5+11+17+ ··· +(6 n - 1) + (6( n +1) - 1) = 3( n +1) 2 +2( n +1) . We have 5+11+17+ ··· +(6 n - 1) + (6( n +1) - 1) = 3 n 2 +2 n + (6( n +1) - 1) =3 n 2 +2 n +6 n +5 =3 n 2 +6 n +3+2 n +2 =3( n +1) 2 +2( n +1) so we are done. (The ±rst equality uses the induction hypothesis.) 2 . Use the Rational Zeros Theorem to prove that b = 3 ± 1+ 2 is irrational. We have b 3 =1+ 2so b 3 - 1= 2 and ( b 3 - 1) 2 = 2, which is b 6 - 2 b 3 +1=2o r b 6 - 2 b 3 - 1 = 0. By the rational zeros theorem, any rational solution of this equation must be of the form b = p q where q divides the coeﬃcient of

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M413test1s - M413 Test 1 Solutions 1 Use induction to prove...

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