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Unformatted text preview: 3. Fix real numbers a , b , c , and d . Prove that ( a + b ) + ( c + d ) = (( d + c ) + a ) + b. 2 MATH 74 MIDTERM 1 Instructions for 4-6. For the remaining three proofs you may make free use of the basic notational conventions of arithmetic (order of operations, no need for various parentheses, etc). As a general guide, anything above the numbered list of “Facts” on the handout, or anything that is one or two steps removed from those, may be assumed and used without explicit reference. If you use a listed fact, however, I would like you to cite it explicitly (in words, or by number). 4. Prove that for any n ∈ N , we have ∑ n j =1 j 3 = n 2 ( n +1) 2 4 . 5. Suppose that x is a rational number and that y is an irrational number. Prove that x + y is irrational. 6. Prove that for any n ∈ N , the number 5 2 n-1 + 1 is an integer multiple of 6....
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- Fall '07
- Math, Rational number, Binary operation