m147A1sol - MATH 147 Assignment #1 Solutions 1) Prove that:...

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Unformatted text preview: MATH 147 Assignment #1 Solutions 1) Prove that: a) 1 2 + 2 2 + + n 2 = n ( n + 1)(2 n + 1) 6 for each n N . Solution: (5 marks). Let P ( n ) be the statement that 1 2 + 2 2 + + n 2 = n ( n + 1)(2 n + 1) 6 . We first show that P (1) holds. But this is true since 1 = 1(1 + 1)(2 1 + 1) 6 = 6 . We now assume that P ( k ) holds. Then 1 2 + 2 2 + + ( k + 1) 2 = (1 2 + 2 2 + + k 2 ) + ( k + 1) 2 . Since P ( k ) holds, we get that 1 2 + 2 2 + + ( k + 1) 2 = (1 2 + 2 2 + + k 2 ) + ( k + 1) 2 = k ( k + 1)(2 k + 1) 6 + k + 1 2 = ( k + 1)( k (2 k + 1) + 6( k + 1)) 6 = ( k + 1)(2 k 2 + k + 6 k + 6) 6 = ( k + 1)(2 k 2 + 7 k + 6) 6 = ( k + 1)( k + 2)(2 k + 3) 6 = ( k + 1)(( k + 1) + 1)(2( k + 1) + 1) 6 1 We have shown that 1 2 + 2 2 + + ( k + 1) 2 = ( k + 1)(( k + 1) + 1)(2( k + 1) + 1) 6 which means that P ( k + 1) holds. Since P (1) holds and we have shown that P ( k ) implies P ( k + 1), we can conclude by induction that P ( n ) holds for all n . b) 2 n + 3 n is divisible by 5 for each odd n N . Solution: (5 marks). Note that the odd numbers are all numbers of the form 2 k- 1 for k N . Let P ( n ) be the statement that 2 2 k- 1 + 3 2 k- 1 is divisible by 5 for each k N . When n = 1, we have 2 2(1)- 1 +3 2(1)- 1 = 2+3 = 5 which is clearly divisible by 5. This hhows that P (1) holds. Assume that P ( j ) holds. That is 2 2 j- 1 + 3 2 j- 1 is divisible by 5. Then 2 2( j +1)- 1 + 3 2( j +1)- 1 = 2 2 j +1 + 3 2 j +1 = 2 2 2 2 j- 1 + 3 2 3 2 j- 1 = 4 2 2 j- 1 + 9 3 2 j- 1 = [4 2 2 j- 1 + 4 3 2 j- 1 ] + 5 3 2 j- 1 = 4 [2 2 j- 1 + 3 2 j- 1 ] + 5 3 2 j- 1 The induction hypothesis tells us that 4 [2 2 j- 1 +3 2 j- 1 ] is divisible by 5. Clearly 5 3 2 j- 1 is divisible by 5. As such P ( j + 1) holds. Using the Principle of Mathematical Induction, we see that P ( k ) holds for all n N . 2) Let a 1 = 1 and for each n 1 let a n +1 = 3 + 2 a n (This is called a recursively defined sequence.) 2 Prove that for every n N , we have a n a n +1 3 Solution: (5 marks). Let P ( n ) be the statement that a n a n +1 3 ....
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m147A1sol - MATH 147 Assignment #1 Solutions 1) Prove that:...

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