review1 - MAT 312/AMS 351 Fall 2010 Review for Midterm 1...

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Unformatted text preview: MAT 312/AMS 351 Fall 2010 Review for Midterm 1 1.2. Understand how to use induction to prove that a statement P ( n ) holds for every integer n . Example: P ( n ) is the statement 1 + 2 + . . . + n = n ( n +1) 2 . Problem 2 p.14. Example: The binomial coefficients parenleftbigg n k parenrightbigg are defined for k n by parenleftbigg n parenrightbigg = parenleftbigg n n parenrightbigg = 1 and parenleftbigg n + 1 k + 1 parenrightbigg = parenleftbigg n k parenrightbigg + parenleftbigg n k + 1 parenrightbigg ; and P ( n ) is the statement that parenleftbigg n k parenrightbigg = n ! k !( n- k )! for every 0 k n . 1.3 Understand the statement of the division algorithm, especially how to show the uniqueness of the quotient and the remainder. Understand the definition of the greatest common divisor d of two positive integers a and b , and the notation d = ( a, b ). Be able to apply the Euclidean Algorithm to two integers a and b , yielding their g.c.d. d . Be able to use that calculation to....
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