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Okanagan College Math Problem of the Month FEBRUARY 2010 – Solutions MAIN PROBLEM: Find one possible value for the next term in this sequence, and explain the pattern: 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, ... Solution: 1705. The sequence is similar to the Fibonacci sequence in that each number is the sum of the previous numbers, but in this sequence it is the sum of the three previous numbers. The next term is the sum of the last three terms given, 274 + 504 + 927, or 1705. BONUS PROBLEM: Consider the sequences 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ... 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, … a) Predict the next two elements in each sequence. b) Find a formula applying to both sequences that gives the nth element in each sequence in terms of one or more earlier elements in the sequence. Solution: a) The next two elements in the first sequence are 6 and 6, while 17 and 19 are the next two elements in the second sequence. b) The formula for both sequences is given by: tn3 tn2 tn1 tn . In words, we add up the two previous values in the sequence and subtract the third previous one from their sum. The differences between the sequences are the values with which you start: In sequence one, 1, 1, 2 are the first three terms, while 1, 2, 4 are the first three terms in the second one.