{"id":149925,"date":"2024-10-08T05:38:10","date_gmt":"2024-10-08T05:38:10","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=149925"},"modified":"2024-10-08T05:38:13","modified_gmt":"2024-10-08T05:38:13","slug":"write-the-recursive-formula-for-the-following-sequence-0-25-0-75-2-25-6-75","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/08\/write-the-recursive-formula-for-the-following-sequence-0-25-0-75-2-25-6-75\/","title":{"rendered":"Write the recursive formula for the following sequence {0.25, 0.75, 2.25, 6.75 \u2026}"},"content":{"rendered":"\n

Write the recursive formula for the following sequence {0.25, 0.75, 2.25, 6.75 \u2026}<\/p>\n\n\n\n

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

The correct answer is: a_1 = 0.25<\/strong><\/mark><\/p>\n\n\n\n

To find the recursive formula for the sequence ({0.25, 0.75, 2.25, 6.75, \\ldots}), we first need to identify the pattern in the sequence and how each term relates to its predecessor.<\/p>\n\n\n\n

Identifying the Pattern<\/h3>\n\n\n\n

Let’s denote the (n)-th term of the sequence as (a_n):<\/p>\n\n\n\n

    \n
  • (a_1 = 0.25)<\/li>\n\n\n\n
  • (a_2 = 0.75)<\/li>\n\n\n\n
  • (a_3 = 2.25)<\/li>\n\n\n\n
  • (a_4 = 6.75)<\/li>\n<\/ul>\n\n\n\n

    Next, we will observe the ratios of successive terms:<\/p>\n\n\n\n

      \n
    • (\\frac{a_2}{a_1} = \\frac{0.75}{0.25} = 3)<\/li>\n\n\n\n
    • (\\frac{a_3}{a_2} = \\frac{2.25}{0.75} = 3)<\/li>\n\n\n\n
    • (\\frac{a_4}{a_3} = \\frac{6.75}{2.25} = 3)<\/li>\n<\/ul>\n\n\n\n

      It appears that each term is multiplied by 3 as we move from one term to the next. However, simply multiplying by 3 doesn’t give us a straightforward recursive relationship. Instead, we can examine the differences between the terms:<\/p>\n\n\n\n

        \n
      • (a_2 – a_1 = 0.75 – 0.25 = 0.50)<\/li>\n\n\n\n
      • (a_3 – a_2 = 2.25 – 0.75 = 1.50)<\/li>\n\n\n\n
      • (a_4 – a_3 = 6.75 – 2.25 = 4.50)<\/li>\n<\/ul>\n\n\n\n

        The differences themselves appear to follow a pattern:<\/p>\n\n\n\n

          \n
        • The differences are (0.50, 1.50, 4.50), which can be rewritten as (0.5 \\cdot 1, 0.5 \\cdot 3, 0.5 \\cdot 9).<\/li>\n<\/ul>\n\n\n\n

          The pattern in the differences seems to indicate that the change from one term to the next involves multiplying the previous term by 3 and then subtracting a value that is also increasing as powers of 3.<\/p>\n\n\n\n

          Recursive Formula<\/h3>\n\n\n\n

          From our observations, we can formulate the recursive equation as follows:<\/p>\n\n\n\n

          [
          a_n = 3a_{n-1} – 0.5 \\cdot 3^{n-2} \\quad \\text{for } n \\geq 2
          ]<\/p>\n\n\n\n

          With the initial term given by:
          [
          a_1 = 0.25
          ]<\/p>\n\n\n\n

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

          In this recursive formula, each term (a_n) is generated by taking three times the previous term (a_{n-1}) and adjusting it by subtracting a scaled value that reflects the rapid growth of the sequence. The presence of (3^{n-2}) indicates that the influence of the previous terms grows exponentially as (n) increases, creating a sequence that accelerates in growth. This understanding helps recognize that while multiplication drives the growth, the subtractive term accounts for the deviations seen in the sequence, providing a complete representation of how each term is generated based on its predecessor.<\/p>\n","protected":false},"excerpt":{"rendered":"

          Write the recursive formula for the following sequence {0.25, 0.75, 2.25, 6.75 \u2026} The Correct answer and Explanation is : The correct answer is: a_1 = 0.25 To find the recursive formula for the sequence ({0.25, 0.75, 2.25, 6.75, \\ldots}), we first need to identify the pattern in the sequence and how each term relates […]<\/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-149925","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149925","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=149925"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149925\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=149925"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=149925"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=149925"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}