4200t1 - n r 3(d By considering the numbers of subsets of...

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Math 4200 Section 2/ Fall 2002 Test 1 Name: Please provide full explanations with your answers. No credit will be given for an answer which is not explained. Your work must be presented in a clean and formal way. 1. Prove by induction that 2 + 4 + · · · + 2 n = n ( n + 1) for every positive integer n . (10pts)
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2. Prove by induction that 4 n + 2 is divisible by 6, for every positive integer n . (10pts)
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3. (a) What is the coefficient of x 5 in the expansion of (1 + x ) 20 ? (4pts) (b) What is the coefficient of x - 14 in the expansion of £ x + x - 1 / 20 ? (4pts)
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3(c) What is the value of the sum n r =0 ( n - 2 r ) ( n r ) ? (4pts) [Hint: Examine part (b) with 20 replaced by n , or use properties of
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Unformatted text preview: ( n r ) .] 3(d) By considering the numbers of subsets of different sizes of a set of n elements, explain why (4pts) ± n ¶ + ± n 1 ¶ + ··· + ± n n ¶ = 2 n 4. Work out the two values of (5pts) √ 1 + i [Use any reasonable method.] 5. (a) Write out the quotient and remainder obtained by dividing 525 by 231. (5pts) (b) Use Euclid’s algorithm to work out the greatest common divisor of 525 and 231. (5pts) 5(c) Express the greatest common divisor of 525 and 231 in the form (5pts) 525 a + 231 b where you must work out the integers a and b ....
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