{"id":230197,"date":"2025-06-09T05:12:26","date_gmt":"2025-06-09T05:12:26","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=230197"},"modified":"2025-06-09T05:12:28","modified_gmt":"2025-06-09T05:12:28","slug":"a-sequence-of-numbers-begins-with-12-and-progresses-geometrically-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/09\/a-sequence-of-numbers-begins-with-12-and-progresses-geometrically-2\/","title":{"rendered":"A sequence of numbers begins with 12 and progresses geometrically."},"content":{"rendered":"\n

A sequence of numbers begins with 12 and progresses geometrically. Each number is the previous number divided by 2 Which value can be used as the common ration in an explicit formula that represents the sequence<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer:<\/h3>\n\n\n\n

The value that can be used as the common ratio<\/strong> in an explicit formula for this sequence is: 12\\boxed{\\frac{1}{2}}21\u200b\u200b<\/p>\n\n\n\n

\n\n\n\n

Explanation <\/h3>\n\n\n\n

A geometric sequence<\/strong> is a sequence of numbers in which each term after the first is found by multiplying the previous one by a constant called the common ratio<\/strong> (usually denoted by rrr).<\/p>\n\n\n\n

In this problem, the sequence begins with 12 and each subsequent term is divided by 2<\/strong>. Mathematically, dividing by 2 is the same as multiplying by 12\\frac{1}{2}21\u200b. So the sequence looks like this: 12, 6, 3, 1.5, 0.75, \u202612,\\ 6,\\ 3,\\ 1.5,\\ 0.75,\\ \\dots12, 6, 3, 1.5, 0.75, \u2026<\/p>\n\n\n\n

To check this, we can divide each term by the previous term:<\/p>\n\n\n\n

    \n
  • 612=12\\frac{6}{12} = \\frac{1}{2}126\u200b=21\u200b<\/li>\n\n\n\n
  • 36=12\\frac{3}{6} = \\frac{1}{2}63\u200b=21\u200b<\/li>\n\n\n\n
  • 1.53=12\\frac{1.5}{3} = \\frac{1}{2}31.5\u200b=21\u200b<\/li>\n<\/ul>\n\n\n\n

    This confirms that the common ratio r=12r = \\frac{1}{2}r=21\u200b.<\/p>\n\n\n\n

    \n\n\n\n

    Explicit Formula<\/h3>\n\n\n\n

    An explicit formula for a geometric sequence is: an=a1\u22c5rn\u22121a_n = a_1 \\cdot r^{n-1}an\u200b=a1\u200b\u22c5rn\u22121<\/p>\n\n\n\n

    Where:<\/p>\n\n\n\n

      \n
    • ana_nan\u200b is the nnn-th term,<\/li>\n\n\n\n
    • a1a_1a1\u200b is the first term of the sequence,<\/li>\n\n\n\n
    • rrr is the common ratio,<\/li>\n\n\n\n
    • nnn is the term number.<\/li>\n<\/ul>\n\n\n\n

      In this case:<\/p>\n\n\n\n

        \n
      • a1=12a_1 = 12a1\u200b=12,<\/li>\n\n\n\n
      • r=12r = \\frac{1}{2}r=21\u200b,<\/li>\n<\/ul>\n\n\n\n

        So the explicit formula becomes: an=12\u22c5(12)n\u22121a_n = 12 \\cdot \\left(\\frac{1}{2}\\right)^{n-1}an\u200b=12\u22c5(21\u200b)n\u22121<\/p>\n\n\n\n

        This formula allows us to find any term in the sequence directly, without computing all the previous terms.<\/p>\n\n\n\n

        \n\n\n\n

        Conclusion<\/h3>\n\n\n\n

        The common ratio in this geometric sequence is 12\\boxed{\\frac{1}{2}}21\u200b\u200b, since each term is found by multiplying the previous term by 12\\frac{1}{2}21\u200b. This common ratio is essential in writing the explicit formula, which models the pattern of the sequence precisely.<\/p>\n\n\n\n

        \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

        A sequence of numbers begins with 12 and progresses geometrically. Each number is the previous number divided by 2 Which value can be used as the common ration in an explicit formula that represents the sequence The Correct Answer and Explanation is: Correct Answer: The value that can be used as the common ratio in […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-230197","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230197","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=230197"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230197\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=230197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=230197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=230197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}