Problem 2 (Periodic Sequence Problem)

Show that the sequence defined by

is periodic.

Solution

This problem is one where you can practice your algebraic skills with radicals. We simply write

Whew! We were just about ready to give up! So, we know that a7 = a₁. Hence, we know the sequence repeats itself every six terms; a₁ = a₇ = a₁₃ = ... . Of course, we also have the relationships a₂ = a₈ = a₁₄ = ... , a3 = a₉ = a15 ... and so on. To make your work simple, you could use a computer algebra system like Maple or Mathematica to carry out all these algebraic calculations. A computer would carry out these computations in a second. If you are not familiar with computer algebra systems, go to the math links on this web site and look up computer algebra systems.

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Last modified on Tuesday, January 12, 1999