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Worksheet: Iterated Functions

Q1:

We can define a sequence π‘₯ , π‘₯ , π‘₯ , … 0 1 2 using the recursive formula π‘₯ = 𝑓 ( π‘₯ ) 𝑖 𝑖 βˆ’ 1 for some function 𝑓 and initial value π‘₯ 0 .

Write the first 5 terms of the sequence generated by 𝑓 ( π‘₯ ) = 8 βˆ’ 3 π‘₯ 𝑖 𝑖 βˆ’ 1 , π‘₯ = βˆ’ 1 1 0 .

  • A βˆ’ 1 1 , βˆ’ 2 5 , 8 3 , βˆ’ 2 4 1 , 7 3 1
  • B βˆ’ 1 1 , 8 , βˆ’ 1 6 , 5 6 , βˆ’ 1 6 0
  • C βˆ’ 1 1 , 4 1 , 1 1 5 , 3 5 3 , 1 0 5 1
  • D βˆ’ 1 1 , 4 1 , βˆ’ 1 1 5 , 3 5 3 , βˆ’ 1 0 5 1
  • E βˆ’ 1 1 , 4 1 , βˆ’ 1 1 5 , βˆ’ 3 3 7 , 1 0 1 9

Q2:

Given an initial value π‘₯ 0 , we can iterate a function using the recursive formula π‘₯ = 𝑓 ( π‘₯ ) 𝑖 𝑖 βˆ’ 1 .

Find π‘₯ 3 given the function 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 2 with initial value π‘₯ = 2 0 .