409Hw10ans - STAT 409 Fall 2011 Homework#10(due Friday...

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STAT 409 Homework #10 Fall 2011 (due Friday, November 18, by 4:00 p.m.) 1. Use mathematical induction to prove that 1 3 + 2 3 + 3 3 + … n 3 = ( ) 4 1 2 2 + n n for all positive integers n . Base. n = 1. 1 3 = ( ) 4 1 1 1 2 2 + checkbld Step. Suppose 1 3 + 2 3 + 3 3 + … k 3 = ( ) 4 1 2 2 + k k . 1 3 + 2 3 + 3 3 + … k 3 + ( k + 1 ) 3 = ( ) 4 1 2 2 + k k + ( k + 1 ) 3 = ( ) + + + 1 4 1 2 2 k k k = ( ) + + + 4 4 4 1 2 2 k k k = ( ) ( ) 4 2 1 2 2 + + k k = ( ) ( ) 4 1 1 1 2 2 + + + k k checkbld 2. Prove that for all positive integers n , n 3 + 2 n is divisible by 3. Base. n = 1. 1 3 + 2 1 = 3 is divisible by 3 checkbld Step. Suppose k 3 + 2 k is divisible by 3. ( k + 1 ) 3 + 2 ( k + 1 ) = k 3 + 3 k 2 + 3 k + 1 + 2 k + 2 = [ k 3 + 2 k ] + 3 [ k 2 + k + 1 ] is divisible by 3 checkbld
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3. Prove that for all positive integers n 4, n ! > 2 n . Base. n = 4. 4 ! = 24 > 16 = 2 4 checkbld Step. Suppose k ! > 2 k for k 4. ( k + 1 ) ! = k ! ( k + 1 ) > 2 k ( k + 1 ) > 2 k 2 = 2 k + 1 checkbld From the textbook: 8th edition ( ) 8.5-4 ( ) Ordered | x – 0 | : 0.0790 0.5901 0.7757 1.0962 1.9415 Rank: 1 2 3 4 5 3.0678 3.8545 5.9848 9.3820 74.0216 Rank: 6 7 8 9 10 W = – 1 + 2 – 3 – 4 – 5 – 6 + 7 – 8 + 9 – 10 =
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