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 + c 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 c 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 c 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 c
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3. Prove that for all positive integers n 4, n ! > 2 n . Base. n = 4. 4 ! = 24 > 16 = 2 4 c Step. Suppose k ! > 2 k for k 4. ( k
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This note was uploaded on 12/15/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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

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