{"id":199419,"date":"2025-03-11T17:47:39","date_gmt":"2025-03-11T17:47:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=199419"},"modified":"2025-03-11T17:47:42","modified_gmt":"2025-03-11T17:47:42","slug":"which-formula-can-be-used-to-describe-the-sequence","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/11\/which-formula-can-be-used-to-describe-the-sequence\/","title":{"rendered":"Which formula can be used to describe the sequence"},"content":{"rendered":"\n
1. Which formula can be used to describe the sequence? -3,(3)\/(5),-(3)\/(25),(3)\/(125),-(3)\/(625) f(x)=-3((1)\/(5))^(x-1) f(x)=-3(-(1)\/(5))^(x-1) f(x)=-(1)\/(5)(3)^(x-1) f(x)=-(1)\/(5)(-3)^(x-1)<\/p>\n\n\n\n
2. Solve. (2+z)(4z+5)=0<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n
1. Identifying the Formula for the Sequence<\/strong><\/h3>\n\n\n\n
The given sequence is: \u22123,35,\u2212325,3125,\u22123625-3, \\frac{3}{5}, -\\frac{3}{25}, \\frac{3}{125}, -\\frac{3}{625}<\/p>\n\n\n\n
Step 1: Identifying the Common Ratio<\/strong><\/h4>\n\n\n\n
Each term alternates in sign and follows a pattern:<\/p>\n\n\n\n
\n
The first term is \u22123-3.<\/li>\n\n\n\n
The second term is obtained by multiplying by \u221215-\\frac{1}{5}: \u22123\u00d7(\u221215)=35-3 \\times \\left(-\\frac{1}{5}\\right) = \\frac{3}{5}<\/li>\n\n\n\n
The third term: 35\u00d7(\u221215)=\u2212325\\frac{3}{5} \\times \\left(-\\frac{1}{5}\\right) = -\\frac{3}{25}<\/li>\n\n\n\n
The fourth term: \u2212325\u00d7(\u221215)=3125-\\frac{3}{25} \\times \\left(-\\frac{1}{5}\\right) = \\frac{3}{125}<\/li>\n\n\n\n
The fifth term: 3125\u00d7(\u221215)=\u22123625\\frac{3}{125} \\times \\left(-\\frac{1}{5}\\right) = -\\frac{3}{625}<\/li>\n<\/ul>\n\n\n\n
Since the ratio between terms is consistently \u221215-\\frac{1}{5}, the general formula follows the geometric sequence formula: f(x)=a\u22c5rx\u22121f(x) = a \\cdot r^{x-1}<\/p>\n\n\n\n
where a=\u22123a = -3 (first term) and r=\u221215r = -\\frac{1}{5}.<\/p>\n\n\n\n
Thus, the correct formula is: f(x)=\u22123(\u221215)x\u22121f(x) = -3 \\left(-\\frac{1}{5}\\right)^{x-1}<\/p>\n\n\n\n
Thus, the two solutions are: z=\u22122orz=\u221254\\mathbf{z = -2 \\quad \\text{or} \\quad z = -\\frac{5}{4}}<\/p>\n\n\n\n
\n\n\n\n
Explanation (300 Words)<\/strong><\/h4>\n\n\n\n
To solve (2+z)(4z+5)=0(2+z)(4z+5) = 0, we use the Zero-Product Property<\/strong>, which states that if the product of two expressions is 0, then at least one of the expressions must be 0.<\/p>\n\n\n\n
We start by setting each factor equal to zero:<\/p>\n\n\n\n
\n
First factor: 2+z=02+z=0<\/strong>\n
\n
Subtracting 2 from both sides: z=\u22122z = -2<\/li>\n<\/ul>\n<\/li>\n\n\n\n
Second factor: 4z+5=04z+5=0<\/strong>\n
\n
Subtracting 5 from both sides: 4z=\u221254z = -5<\/li>\n\n\n\n
Dividing by 4: z=\u221254z = -\\frac{5}{4}<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
Thus, the equation has two solutions: z=\u22122z = -2 and z=\u221254z = -\\frac{5}{4}.<\/p>\n\n\n\n
Why is this important?<\/strong><\/h4>\n\n\n\n
This method is widely used in algebra when solving quadratic or polynomial equations. Recognizing factors and using the Zero-Product Property allows us to find solutions efficiently without expanding and solving a full quadratic equation.<\/p>\n\n\n\n
Additionally, this problem shows how factoring helps break complex expressions into simpler parts. If this equation were expanded, it would be a quadratic equation that we would need to solve using the quadratic formula or factoring techniques.<\/p>\n\n\n\n
Understanding this process helps build algebraic skills, which are essential in higher mathematics and real-world applications such as physics, engineering, and economics.<\/p>\n\n\n\n
Now, let me generate an image illustrating this solution.<\/p>\n\n\n\n
Here’s a graphical representation of the equation (2+z)(4z+5)=0(2+z)(4z+5)=0. The red points indicate the solutions z=\u22122z = -2 and z=\u221254z = -\\frac{5}{4}, where the function intersects the x-axis. Let me know if you need any further explanation!<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
1. Which formula can be used to describe the sequence? -3,(3)\/(5),-(3)\/(25),(3)\/(125),-(3)\/(625) f(x)=-3((1)\/(5))^(x-1) f(x)=-3(-(1)\/(5))^(x-1) f(x)=-(1)\/(5)(3)^(x-1) f(x)=-(1)\/(5)(-3)^(x-1) 2. Solve. (2+z)(4z+5)=0 The correct answer and explanation is: 1. Identifying the Formula for the Sequence The given sequence is: \u22123,35,\u2212325,3125,\u22123625-3, \\frac{3}{5}, -\\frac{3}{25}, \\frac{3}{125}, -\\frac{3}{625} Step 1: Identifying the Common Ratio Each term alternates in sign and follows a pattern: […]<\/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-199419","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/199419","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=199419"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/199419\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=199419"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=199419"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=199419"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-862-1024x705.png\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"