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Finding a Pattern Notes - Sequences -The study of ________________ is often called the study of _________________. -As a problem-solving strategy, _____________ patterns enable you to ____________ a complex problem to a _________________ and then use the pattern to find a _____________. Often the key to finding a pattern is _______________ information. A) A __________________ is an ordered string of numbers tied together by a ___________ ___________, or set of rules, that _________________ the next term in the sequence. -A ________________ is an individual member of a sequence. Ex) Find the pattern and predict the next four terms. Then write a sentence to explain the pattern. 1. 1, 2, 4, _______, _______, _______, ________ a. _______________ each term to find the next term in the sequence. b. Start by _________________ one to the first term, then _________ one greater number to each successive term. 2. 1, 3, 5, 7, _______, _______, ________, ________ a. ________ _________ to each term to find the next. 3. 1, 6, 11, 16, ________, ________, _________, ________ a. _________ ________ to each term. 4. 1, 3, 6, 10, ________, _______, ________, ________ a. Start by _________ _______ to the first term, then _______ ________ greater number to each successive term. Having trouble with sequences? Try the following process: 1) _______________ down the sequence you’re working with: 3,4,6,9,13,18,24,… 2) Write down each __________ of the sequence and its _______________ in the sequence, as show in the table below. Position Term 1 3 2 4 3 6 4 9 5 13 6 18 7 24 6 18 7 24 3) Write down the _________________ between the terms of the sequence: Position Term 1 3 2 4 3 6 4 9 5 13 4) ________________ for a way to ___________ the terms to their positions. Look for a way to relate the ___________________ between terms to the ___________________. Position Term 1 3 2 4 3 6 4 9 5 13 6 18 7 24 5) If figuring the differences between terms doesn’t work, try using other operations. (mult., div,..) One __________ type of sequence is called the _____________ Sequence. This sequence has __________ to start it off and usually it’s the first ______ numbers, sometimes even the first _________. This sequence _________ the two or three __________ terms to find the next. Find the next two terms Ex) 2, 2, 4, 6, 10, 16, 26, ______, ______ Ex) 1, 2, 3, 6, 11, 20, 37, ______, ______ Patterns can be identified on different level of ___________________. That is, sometimes it lies in the differences between the ______________, sometimes it lies in the differences between the ________________ and so on. Finding Patterns Example Dodger Stadium Radio broadcasters joke about the number of people who start leaving Dodger Stadium during the 7th inning of baseball games. One evening, during a particularly boring baseball game in which the Dodgers were trailing by six runs after six innings, the fans began to leave at a record pace. After the first out in the top of the seventh inning, 100 fans left. After the 2nd out, 150 fans left. After the third out, 200 fans left. The pattern continued in this way, with 50 more fans leaving after each out than had left after the previous out. The ridiculous thing was, the Dodgers tied up the game in the bottom of the 9th inning, and people still kept leaving early. The game lasted 10 innings, and the pattern continued through the bottom of the 10th inning. How many fans left early? Out # 1 2 3 4 5 Base Fans Leaving Additional Fans Leaving Additional Fans Leaving Additional Fans Leaving 6 7 … Total … … … … Pattern: 2nd Paragraph I solved this problem by ___________ a ________________. First, I made a ______________ with the outs and total labeled on top and the number of __________ leaving on the left. Next, I knew the base number of fans leaving every inning so I filled in 100 in every column of that row. Then, I added another ________ fans every inning of play ______________ the columns of outs that was before the out number I was on. After that, I saw a _____________ in the total number column after the first 2400 so I did not make the chart for the total 24 outs nor did I continue with the additional fans leaving rows. Lastly, I followed the pattern and added the _____________ number column to find the total number of fans leaving. The total number of fans leaving was ________________ people. Finding Pattern Notes - Functions Sequences taken from algebra are featured next. These problems are models of ________________. A function assigns an ___________ to each _____________. For a function there is a consistent rule that uses the input number to generate the output number. Ex) Table of values – Determine the rule for each function shown below In x 0 1 2 3 4 5 895 Out ? 5 6 7 8 9 ? ? To find each successive terms, you’d simply add 1 to each previous term. However looking at this example in the form of a table of values allows you to find a _____________ that you can apply to a certain term to find every other term. Let x represent that certain term, or input. The table shows that when x is 2, the output is _____. When x is 4, the output is ______. So the rule of __________ _____ to the input seems to work for the inputs 2 and 4. Experimenting further shows you that this rule works for all the inputs. Now that we know that x+5 is the function rule to apply to each input, determining the output for an input of 895 is easy: 895+5=900. *If you can’t figure out what the rules are, simply treat each problem as if it was a sequence written _________________. Clearly there is a big difference between finding patterns in tables of values and finding patterns in simple sequences. *When finding patterns in _________________, you attempt to go through the entire sequence, always try the next term in the sequence. *When looking for a pattern in a ____________ of ______________ , you try to find a relationship between the input number and the output number. -Finding this _______________ allows you to jump way ahead in the sequence and find the _____________ number (such as 895) for a ___________ number _________________ having to extend the sequence all the way to the 895th term.