Geometric Sequences: Apply It

  • Find the common ratio for a geometric sequence.
  • Write the formula for a geometric sequence.
  • Use geometric sequences to solve realistic scenarios

Dr. Chen prescribes a new antibiotic for Sofia, who has developed a bacterial infection. The medication needs to maintain a certain concentration in her bloodstream to be effective, but her body naturally eliminates the drug over time through her liver and kidneys.

Initial Dosage Analysis

Sofia takes her first 800mg dose of the antibiotic at 8:00 AM. Dr. Chen explains that her body eliminates 50% of the medication from her bloodstream every 6 hours.

The explicit form of a geometric sequence is [latex]a_n=a_1(r)^{n}[/latex]

For geometric sequences that change every [latex]t[/latex] hours we can adjust the explicit formula to be:

[latex]a_n=a_1(r)^{\frac{n}{t}}[/latex]

The antibiotic is only effective when at least 200 mg remains in Sofia’s bloodstream. Dr. Chen needs to determine how long the medication stays above this threshold.