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A sequence is a list of numbers, shapes, letters, etc. that are in a distinct or recognizable pattern. Each number in a sequence is called a term.Recursion or recursive sequence - the value of a term is based on the value of the previous term(s).Eg. Fibonacci 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ……. To find the value of one term, add the previous two terms togethert5=t3+t4t1=1, t2=1, tn=tn-2+tn-1Alternatively, you could write this in function notation: Continuous and Discrete FunctionsContinuous Functions – domain is ____________Example: y=x2Discrete Functions – domain is _____________Example: fn=n2Example 1Write the first 4 terms of each sequence, n∈N(natural numbers)a) 1=5 , fn=fn-1-4b) t1=3, tn=2tn-1-nExample 2Determine a recursion formula fora) -2, 7, 16, 25, …Use tn notation:Use fn notation:b) 1, -3, 9, -27Use tn notation:Use fn notation:Don’t forget you need a starting value when you define your recursion formula! Example 3Find the recursive formula for:2, 4, 6, 8, 10, ____, ____, ____Find an explicit formula for the same sequence (a formula that does not depend on the previous terms). Hint: Can you graph the sequence in some way?Example 4Determine a recursion formula and an explicit formula for each sequence.2, 5, 8, 11, …3, 2, 1, 0, …2, 6, 18, 54…11, 7, 3, -1-3, 6, -12, 24Example 5 Write the first five terms of each sequence. Write an explicit formula for the sequence if possible.t1=11,tn=tn-1-4t1=4, tn=-3tn-1t1=2, tn=tn-1+2n-3t1=5, t2=7, tn=tn-1+tn-2Example 6 A famous recursive sequence is the Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21Write the recursion formula for this sequence.The explicit formula for the Fibonacci Sequence is much more complicated:ChallengeDetermine a recursive formula for the sequence 5, 9, 18, 34, 59, … ................
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