A sequence has a common ratio of Three-halves and f(5) = 81. Which explicit formula represents the sequence? f(x) = 24(Three-halves) Superscript x minus 1 f(x) = 16(Three-halves) Superscript x minus 1 f(x) = 24(Three-halves) Superscript x f(x) = 16(Three-halves) Superscript x<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n The given problem describes a geometric sequence with a common ratio of ( \\frac{3}{2} ) and ( f(5) = 81 ). To find the explicit formula for the sequence, we must first recall the general form of the explicit formula for a geometric sequence, which is:<\/p>\n\n\n\n [ Where:<\/p>\n\n\n\n The problem states that the common ratio ( r ) is ( \\frac{3}{2} ). This means that each term is multiplied by ( \\frac{3}{2} ) to obtain the next term.<\/p>\n\n\n\n We are also told that ( f(5) = 81 ). To find the value of the first term ( a ), we can substitute ( f(5) = 81 ) into the general formula:<\/p>\n\n\n\n [ This simplifies to:<\/p>\n\n\n\n [ Now, calculate ( \\left( \\frac{3}{2} \\right)^4 ):<\/p>\n\n\n\n [ So the equation becomes:<\/p>\n\n\n\n [ To solve for ( a ), multiply both sides of the equation by ( \\frac{16}{81} ):<\/p>\n\n\n\n [ Thus, the first term of the sequence is ( a = 16 ).<\/p>\n\n\n\n Now that we know the first term ( a = 16 ) and the common ratio ( r = \\frac{3}{2} ), we can substitute these values into the general formula:<\/p>\n\n\n\n [ The explicit formula for the sequence is:<\/p>\n\n\n\n [ This matches the second option in the problem. Therefore, the correct answer is:<\/p>\n\n\n\n [ A sequence has a common ratio of Three-halves and f(5) = 81. Which explicit formula represents the sequence? f(x) = 24(Three-halves) Superscript x minus 1 f(x) = 16(Three-halves) Superscript x minus 1 f(x) = 24(Three-halves) Superscript x f(x) = 16(Three-halves) Superscript x The Correct Answer and Explanation is: The given problem describes a geometric sequence […]<\/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-166453","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/166453","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=166453"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/166453\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=166453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=166453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=166453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
f(x) = a \\cdot r^{x – 1}
]<\/p>\n\n\n\n\n
Step 1: Identify the common ratio and the form of the sequence<\/h3>\n\n\n\n
Step 2: Use the given value of ( f(5) )<\/h3>\n\n\n\n
f(5) = a \\cdot \\left( \\frac{3}{2} \\right)^{5 – 1}
]<\/p>\n\n\n\n
81 = a \\cdot \\left( \\frac{3}{2} \\right)^4
]<\/p>\n\n\n\n
\\left( \\frac{3}{2} \\right)^4 = \\frac{3^4}{2^4} = \\frac{81}{16}
]<\/p>\n\n\n\n
81 = a \\cdot \\frac{81}{16}
]<\/p>\n\n\n\nStep 3: Solve for ( a )<\/h3>\n\n\n\n
a = 81 \\cdot \\frac{16}{81} = 16
]<\/p>\n\n\n\nStep 4: Write the explicit formula<\/h3>\n\n\n\n
f(x) = 16 \\cdot \\left( \\frac{3}{2} \\right)^{x – 1}
]<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n
f(x) = 16 \\cdot \\left( \\frac{3}{2} \\right)^{x – 1}
]<\/p>\n\n\n\n
f(x) = 16 \\cdot \\left( \\frac{3}{2} \\right)^{x – 1}
]<\/p>\n","protected":false},"excerpt":{"rendered":"