{"id":3206,"date":"2025-08-15T22:38:09","date_gmt":"2025-08-15T22:38:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3206"},"modified":"2025-09-25T17:42:05","modified_gmt":"2025-09-25T17:42:05","slug":"geometric-sequences-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/geometric-sequences-apply-it\/","title":{"raw":"Geometric Sequences: Apply It","rendered":"Geometric Sequences: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find the common ratio for a geometric sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Write the formula for a geometric sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use geometric sequences to solve realistic scenarios<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<p class=\"whitespace-normal break-words\">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.<\/p>\r\n\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Initial Dosage Analysis<\/h2>\r\n<p class=\"whitespace-normal break-words\">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.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The explicit form of a geometric sequence is [latex]a_n=a_1(r)^{n}[\/latex]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">For geometric sequences that change every [latex]t[\/latex] hours we can adjust the explicit formula to be:\r\n<p style=\"text-align: center;\">[latex]a_n=a_1(r)^{\\frac{n}{t}}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312738[\/ohm_question]\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">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.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312740[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Find the common ratio for a geometric sequence.<\/li>\n<li style=\"font-weight: 400;\">Write the formula for a geometric sequence.<\/li>\n<li style=\"font-weight: 400;\">Use geometric sequences to solve realistic scenarios<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<p class=\"whitespace-normal break-words\">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.<\/p>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Initial Dosage Analysis<\/h2>\n<p class=\"whitespace-normal break-words\">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.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">The explicit form of a geometric sequence is [latex]a_n=a_1(r)^{n}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">For geometric sequences that change every [latex]t[\/latex] hours we can adjust the explicit formula to be:<\/p>\n<p style=\"text-align: center;\">[latex]a_n=a_1(r)^{\\frac{n}{t}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312738\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312738&theme=lumen&iframe_resize_id=ohm312738&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">The antibiotic is only effective when at least 200 mg remains in Sofia&#8217;s bloodstream. Dr. Chen needs to determine how long the medication stays above this threshold.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312740\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312740&theme=lumen&iframe_resize_id=ohm312740&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":18,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":156,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3206"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3206\/revisions"}],"predecessor-version":[{"id":4432,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3206\/revisions\/4432"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/156"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3206\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3206"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3206"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3206"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}