Project 2: Number Patterns, p. 3 Math 214 Project 2. Number Patterns 1. a.) Goldbach's conjecture states that every even number greater than two is the sum of two prime numbers. Make your own example (different from your group) that shows this is true. Share your example with your group and help check that each of your group members is correct! b.) What does conjecture mean? (Do a web search or look in the textbook!) Why is the above called a conjecture? Answer both parts of this question! c.) There is a conjecture called \"Goldbach's weak conjecture\" (ves, really!) that every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). Find two examples of this. Stepl: Understand the problem. Show your word by word translation. Every odd number (make a list) greater than 5 (fix the above list!) can be expressed as the sum of (what does sum mean?) three primes (make a list of primes) Pick your own odd number (different from your group) and show the conjecture is true for that number. Share your work with your group. Project 2: Number Patterns, p. 9 7. Find the pattern when the squares of the Fibonacci numbers are added: a. Complete the table for the first six rows: The squares of Fibonacci Numbers Sum Pattern (F1)? =12 = 1 =1x1 (F1)2 + (F2)? =12+ 1% = 1+1 = 2 =1x2 (F)2+ (F22+ (Fa)2 =12+12+22 = 1+1+4 = 6 =2 x_? = = _ :__x - _ _ _ _ _____X - - = = X Complete the blanks and the next three rows. Hint: look for two special numbers that multiply to get the sum. b. Explain the pattern of the answers in words. Hint: relate the two multiplied numbers to the numbers you just added. c. Use the pattern to predict the sum when 12+ 12+ 22+ 32+ 52 + .., + 2332 1s added. Your answer should show that you know how to get the answer using the pattern, without having to actually add up all the numbers! Hint: which subscript does 233 have? Which subscript will the two numbers in your answer have? How do you know? Project 2: Number Patterns, p. 15 14. In about the third grade, children can work on creating block patterns like L's and staircases to explore the patterns they find. a.) Draw the 5 L-shape and fill in the correct number of blocks for the 4th and 5th shapes. Careful notice how much higher and how much longer the L gets each time! Li=1 Lo=4 Ls=7 L4=? Ls=? b.) Use the pattern of successive differences to find the number of blocks 1n next L's. YU c.) If you graph the number of blocks, will you have a line or a curve? Complete the table and draw the graph. L Number, | Number of y X blocks, y 1 1 2 3 4 b If your graphing skills are a little rusty, check the online grapher at https://www.desmos.com/calculator d.) Conclusion: I have a line/curve (circle the correct one) because.... (think of how many rows of differences you had and look at section 2.4 p. 2 & 3)