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Year 12 Problem-Solving Summer School Problem Sheet 2 – Problems based on number Warm-up! Question 1 Show that, for any natural number n, n(n+1) is even Solution Question 2 For what value of n is n! equal to the exact number of seconds in six weeks? Solution Year 12 Problem-Solving Summer School Question 3 In a sequence of numbers the nth term is given by n2 + 2n. (a) Write the first six numbers in the sequence; (b) Explain why the numbers in the sequence alternates between odd and even. Solution (a) (b) Year 12 Problem-Solving Summer School Question 4 The number (102 – 82)(92 – 72)(82 – 62)(72 – 52)(62 – 42)(52 – 32)(42 – 22)(32 – 12) can be written in the form k × 2n, where k and n are positive integers and k is odd. What are the values of k and n? Solution Year 12 Problem-Solving Summer School Questions to be considered in the next online session Question 1 Five numbers are arranged in order from least to greatest: x, x3, x4, x2, x0 Where does –x–1 belong in the list above? Solution Year 12 Problem-Solving Summer School Question 2 Is n2 + n + 41 a prime number for all natural numbers n? If your answer is yes, prove it. If your answer is no, give a counter-example. Solution Year 12 Problem-Solving Summer School Question 3 How many two digit numbers N have the property that the sum of N and the number formed by reversing the digits of N is a square? Solution Year 12 Problem-Solving Summer School Question 4 Show that, for any natural number n, n(n +1)(2n + 1) is a multiple of 6 Solution Year 12 Problem-Solving Summer School Question 4a Show that every cube number is either a multiple of 9, is one more than a multiple of 9 or is one less than a multiple of 9. Solution Year 12 Problem-Solving Summer School Question 5 The expressions ௫ାଷ ଷ and ଶ௫ିଵ ହ represent a pair of consecutive whole numbers. The expressions ଶ௬ିଵ ହ and whole numbers. What is the value of |x – y|? Solution ୷ାଷ ଷ represent a pair of consecutive Year 12 Problem-Solving Summer School Question 6 (AEA, part) Define f(x) = x – [x], x ≥ 0 where [x] is the largest integer that is less than or equal to x. For example, f(3.7) = 3.7 – 3 = 0.7; f(3) = 3 – 3 = 0. (a) Sketch the graph of y = f(x) for 0 ≤ x < 4. (b) Find the value of p for which 0 fሺxሻdx= 0.18 Solution (a) (b) p Year 12 Problem-Solving Summer School Question 7 (STEP, part) How many integers greater than or equal to zero and less than a million are not divisible by 2 or 5? What is the average value of these integers? Solution