Thursday, October 21, 2010
A Recursively Defined Sequence to Challenge Your Algebra Students
In continued tribute to Dr. Mandelbrot, here is a challenge problem for your Algebra 2 students which develops the ideas of iteration and recursively-defined sequences while providing technical skill practice. From my own experience, even some of the strongest will trip over the details so don't be surprised if you get many different answers for the 5th term in part (c) below! We all know that current texts do not provide enough mechanical practice and this becomes more evident as our top students move into the advanced classes.
THE CHALLENGE
A sequence is defined as follows. Each term after the first is two less than three times the preceding term.
(a) If the first term is 2, determine the 2nd through 5th terms.
(b) If the first term is 1, determine the 100th term. Explain.
(c) If the first term is x, determine simplified expressions in terms of x for the 2nd through 5th terms. To help you verify your answers, the 5th term is 81x - 80. Show all steps clearly. Compare your results with others in your group and resolve any discrepancies.
(d) Write a general expression for the nth term if the 1st term is x. It should work for all terms including the first! Explain your method. Proving your formula works for all n is optional.
Answer: 3^(n-1)x - (3^(n-1) - 1)
NOTE: Students who have learned the formula for the nth term of a geometric sequence should recognize the first term in this answer! Help them to make the connection...
(e) Extension: Change the recursive relationship to: Each term after the first is three less than twice the preceding term. Redo part (d) for this new sequence. The pattern is more challenging!
Ans: 2^(n-1)x - 3(2^(n-1) - 1)
NOTE: For the more advanced students, have them prove their "formula" by induction.
Final Comment: In what form do you think this kind of question would appear on the SATs and, yes, this topic is tested and has appeared!
"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)
"You've got to be taught To hate and fear, You've got to be taught from year to year, It's got to be drummed In your dear little ear. You've got to be carefully taught." --from South Pacific
Posted by Dave Marain at 2:22 PM 4 comments
Labels: recursion, recursively defined sequences, SAT-type problems