Unformatted text preview: r ) be written as n 4 = 3 k + 1 for some integer k . 4. [8 points] Prove that √ 7 is irrational. Hint : Use a proof by contradiction. Also recall that an integer m is divisible by 7 means that we can write m = 7 q for some integer q . 5. [8 points] Use induction to prove that 1 3 + 2 3 + 3 3 + ··· + n 3 = n 2 ( n + 1) 2 4 for all integers n ≥ 1. Remark : Since n 2 ( n +1) 2 4 = h n ( n +1) 2 i 2 , and we have learned that ∑ n i =1 i = n ( n +1) 2 , Problem 5 leads to a fascinating result n X i =1 i 3 = ˆ n X i =1 i ! 2 . 6. [8 points] Use induction to prove that n X i =1 (2 i + 5) = n 2 + 6 n for all integers n ≥ 1. 7. [8 points] Use induction to prove that 3 + 3 · 5 + 3 · 5 2 + ··· + 3 · 5 n1 = 3(5 n1) 4 for all integers n ≥ 1. Suggestion : Instead of using sigma notation, it may be easier to write out the full equation....
View
Full Document
 Spring '08
 Cox
 Math, Division, Remainder, Natural number, 3k, integer k.

Click to edit the document details