{"id":218907,"date":"2025-05-24T16:24:39","date_gmt":"2025-05-24T16:24:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=218907"},"modified":"2025-05-24T16:24:42","modified_gmt":"2025-05-24T16:24:42","slug":"the-first-harmonic-number-is-1-the-second-is-1-5-the-third-is-1-83333-and-so-on","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/24\/the-first-harmonic-number-is-1-the-second-is-1-5-the-third-is-1-83333-and-so-on\/","title":{"rendered":"the first harmonic number is 1, the second is 1.5, the third is 1.83333\u2026 and so on."},"content":{"rendered":"\n

In mathematics, the Nth harmonic number is defined to be 1 + 1\/2 + 1\/3 + 1\/4 + \u2026 + 1\/N. So, the first harmonic number is 1, the second is 1.5, the third is 1.83333\u2026 and so on. Assume that n is an integer variable whose value is some positive integer N. Assume also that hn is a variable whose value is the Nth harmonic number. Write an expression whose value is the (N+1)th harmonic number.<\/p>\n\n\n\n

Python<\/p>\n\n\n\n

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

\u2705 Correct Python Expression:<\/h3>\n\n\n\n
hn_plus_1 = hn + 1 \/ (n + 1)<\/code><\/pre>\n\n\n\n
\n\n\n\n
\ud83e\udde0 Explanation:<\/h3>\n\n\n\n
In mathematics, harmonic numbers<\/strong> are a sequence of numbers formed by taking the sum of the reciprocals of the first n<\/em> positive integers. The Nth<\/em> harmonic number is denoted as:<\/p>\n\n\n\n
$$
H_n = 1 + \\frac{1}{2} + \\frac{1}{3} + \\dots + \\frac{1}{n}
$$<\/p>\n\n\n\n
So, the harmonic number increases as n<\/em> increases, and it\u2019s always the sum of all reciprocal values from 1 to n<\/em>. In programming (e.g., Python), we often need to compute the next value in the sequence without recalculating everything from scratch.<\/p>\n\n\n\n
If you already know the Nth harmonic number<\/strong> hn<\/code> (i.e., the sum up to 1\/n), then to compute the (N+1)th harmonic number<\/strong>, you simply add the next term in the sequence:<\/p>\n\n\n\n
$$
H_{n+1} = H_n + \\frac{1}{n+1}
$$<\/p>\n\n\n\n
This is the core logic behind the Python expression:<\/p>\n\n\n\n
hn_plus_1 = hn + 1 \/ (n + 1)<\/code><\/pre>\n\n\n\n
    \n
  • hn<\/code> holds the value of the current (Nth) harmonic number.<\/li>\n\n\n\n
  • n<\/code> is the integer representing the current value of N<\/em>.<\/li>\n\n\n\n
  • 1 \/ (n + 1)<\/code> computes the next term to be added.<\/li>\n\n\n\n
  • Adding it to hn<\/code> gives you the harmonic number for the next integer (N+1).<\/li>\n<\/ul>\n\n\n\n
    This approach is efficient because it avoids recomputing all the prior terms of the harmonic series. It builds on previous work \u2014 a classic technique in mathematics and programming known as dynamic programming<\/strong> or incremental computation<\/strong>.<\/p>\n\n\n\n
    Such expressions are useful in numerical methods, approximation theory, and algorithms, especially those involving complexity analysis or estimates based on harmonic growth (common in computer science, like in amortized analysis of data structures).<\/p>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
    In mathematics, the Nth harmonic number is defined to be 1 + 1\/2 + 1\/3 + 1\/4 + \u2026 + 1\/N. So, the first harmonic number is 1, the second is 1.5, the third is 1.83333\u2026 and so on. Assume that n is an integer variable whose value is some positive integer N. Assume also […]<\/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-218907","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218907","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=218907"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218907\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=218907"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=218907"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=218907"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}