{"id":90181,"date":"2025-06-01T10:22:25","date_gmt":"2025-06-01T10:22:25","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90181"},"modified":"2025-06-01T10:22:25","modified_gmt":"2025-06-01T10:22:25","slug":"if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/","title":{"rendered":"If the product of $n$ positive numbers is unity, then their sum is"},"content":{"rendered":"

If the product of $n$ positive numbers is unity, then their sum is<\/p>\n

[amp_mcq option1=”a positive integer” option2=”divisible by $n$” option3=”equal to $n + \\frac{1}{n}$” option4=”never less than $n$” correct=”option4″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2017<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is D) never less than n.
\n<\/section>\n
\nLet the n positive numbers be $x_1, x_2, …, x_n$.
\nWe are given that their product is unity: $x_1 \\times x_2 \\times \\dots \\times x_n = 1$.
\nWe want to determine the property of their sum: $S = x_1 + x_2 + \\dots + x_n$.
\nAccording to the Arithmetic Mean – Geometric Mean (AM-GM) inequality, for a set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. Since the numbers are positive, this inequality applies:
\n$\\frac{x_1 + x_2 + \\dots + x_n}{n} \\ge \\sqrt[n]{x_1 x_2 \\dots x_n}$
\nSubstitute the given product into the inequality:
\n$\\frac{S}{n} \\ge \\sqrt[n]{1}$
\n$\\frac{S}{n} \\ge 1$
\n$S \\ge n$
\nThe sum of the n positive numbers whose product is unity is always greater than or equal to n.
\nEquality holds if and only if all the numbers are equal. If $x_1 = x_2 = \\dots = x_n = x$, and their product is 1, then $x^n = 1$. Since they are positive, $x$ must be 1. In this case, the sum is $n \\times 1 = n$. If the numbers are not all equal, the sum is strictly greater than n. Therefore, the sum is never less than n.
\n<\/section>\n
\nThe AM-GM inequality is a fundamental concept in mathematics often used to find minimum or maximum values or to prove inequalities. It states that for non-negative numbers $a_1, a_2, \\dots, a_n$, $\\frac{a_1 + a_2 + \\dots + a_n}{n} \\ge \\sqrt[n]{a_1 a_2 \\dots a_n}$, with equality holding if and only if $a_1 = a_2 = \\dots = a_n$.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

If the product of $n$ positive numbers is unity, then their sum is [amp_mcq option1=”a positive integer” option2=”divisible by $n$” option3=”equal to $n + \\frac{1}{n}$” option4=”never less than $n$” correct=”option4″] This question was previously asked in UPSC CAPF – 2017 Download PDFAttempt Online The correct answer is D) never less than n. Let the n … <\/p>\n

Detailed SolutionIf the product of $n$ positive numbers is unity, then their sum is<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1101,1102],"class_list":["post-90181","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1101","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nIf the product of $n$ positive numbers is unity, then their sum is<\/title>\n<meta name=\"description\" content=\"The correct answer is D) never less than n. Let the n positive numbers be $x_1, x_2, ..., x_n$. We are given that their product is unity: $x_1 times x_2 times dots times x_n = 1$. We want to determine the property of their sum: $S = x_1 + x_2 + dots + x_n$. According to the Arithmetic Mean - Geometric Mean (AM-GM) inequality, for a set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. Since the numbers are positive, this inequality applies: $frac{x_1 + x_2 + dots + x_n}{n} ge sqrt[n]{x_1 x_2 dots x_n}$ Substitute the given product into the inequality: $frac{S}{n} ge sqrt[n]{1}$ $frac{S}{n} ge 1$ $S ge n$ The sum of the n positive numbers whose product is unity is always greater than or equal to n. Equality holds if and only if all the numbers are equal. If $x_1 = x_2 = dots = x_n = x$, and their product is 1, then $x^n = 1$. Since they are positive, $x$ must be 1. In this case, the sum is $n times 1 = n$. If the numbers are not all equal, the sum is strictly greater than n. Therefore, the sum is never less than n.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If the product of $n$ positive numbers is unity, then their sum is\" \/>\n<meta property=\"og:description\" content=\"The correct answer is D) never less than n. Let the n positive numbers be $x_1, x_2, ..., x_n$. We are given that their product is unity: $x_1 times x_2 times dots times x_n = 1$. We want to determine the property of their sum: $S = x_1 + x_2 + dots + x_n$. According to the Arithmetic Mean - Geometric Mean (AM-GM) inequality, for a set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. Since the numbers are positive, this inequality applies: $frac{x_1 + x_2 + dots + x_n}{n} ge sqrt[n]{x_1 x_2 dots x_n}$ Substitute the given product into the inequality: $frac{S}{n} ge sqrt[n]{1}$ $frac{S}{n} ge 1$ $S ge n$ The sum of the n positive numbers whose product is unity is always greater than or equal to n. Equality holds if and only if all the numbers are equal. If $x_1 = x_2 = dots = x_n = x$, and their product is 1, then $x^n = 1$. Since they are positive, $x$ must be 1. In this case, the sum is $n times 1 = n$. If the numbers are not all equal, the sum is strictly greater than n. Therefore, the sum is never less than n.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:22:25+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"If the product of $n$ positive numbers is unity, then their sum is","description":"The correct answer is D) never less than n. Let the n positive numbers be $x_1, x_2, ..., x_n$. We are given that their product is unity: $x_1 times x_2 times dots times x_n = 1$. We want to determine the property of their sum: $S = x_1 + x_2 + dots + x_n$. According to the Arithmetic Mean - Geometric Mean (AM-GM) inequality, for a set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. Since the numbers are positive, this inequality applies: $frac{x_1 + x_2 + dots + x_n}{n} ge sqrt[n]{x_1 x_2 dots x_n}$ Substitute the given product into the inequality: $frac{S}{n} ge sqrt[n]{1}$ $frac{S}{n} ge 1$ $S ge n$ The sum of the n positive numbers whose product is unity is always greater than or equal to n. Equality holds if and only if all the numbers are equal. If $x_1 = x_2 = dots = x_n = x$, and their product is 1, then $x^n = 1$. Since they are positive, $x$ must be 1. In this case, the sum is $n times 1 = n$. If the numbers are not all equal, the sum is strictly greater than n. Therefore, the sum is never less than n.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/","og_locale":"en_US","og_type":"article","og_title":"If the product of $n$ positive numbers is unity, then their sum is","og_description":"The correct answer is D) never less than n. Let the n positive numbers be $x_1, x_2, ..., x_n$. We are given that their product is unity: $x_1 times x_2 times dots times x_n = 1$. We want to determine the property of their sum: $S = x_1 + x_2 + dots + x_n$. According to the Arithmetic Mean - Geometric Mean (AM-GM) inequality, for a set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. Since the numbers are positive, this inequality applies: $frac{x_1 + x_2 + dots + x_n}{n} ge sqrt[n]{x_1 x_2 dots x_n}$ Substitute the given product into the inequality: $frac{S}{n} ge sqrt[n]{1}$ $frac{S}{n} ge 1$ $S ge n$ The sum of the n positive numbers whose product is unity is always greater than or equal to n. Equality holds if and only if all the numbers are equal. If $x_1 = x_2 = dots = x_n = x$, and their product is 1, then $x^n = 1$. Since they are positive, $x$ must be 1. In this case, the sum is $n times 1 = n$. If the numbers are not all equal, the sum is strictly greater than n. Therefore, the sum is never less than n.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:22:25+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/","url":"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/","name":"If the product of $n$ positive numbers is unity, then their sum is","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:22:25+00:00","dateModified":"2025-06-01T10:22:25+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is D) never less than n. Let the n positive numbers be $x_1, x_2, ..., x_n$. We are given that their product is unity: $x_1 \\times x_2 \\times \\dots \\times x_n = 1$. We want to determine the property of their sum: $S = x_1 + x_2 + \\dots + x_n$. According to the Arithmetic Mean - Geometric Mean (AM-GM) inequality, for a set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. Since the numbers are positive, this inequality applies: $\\frac{x_1 + x_2 + \\dots + x_n}{n} \\ge \\sqrt[n]{x_1 x_2 \\dots x_n}$ Substitute the given product into the inequality: $\\frac{S}{n} \\ge \\sqrt[n]{1}$ $\\frac{S}{n} \\ge 1$ $S \\ge n$ The sum of the n positive numbers whose product is unity is always greater than or equal to n. Equality holds if and only if all the numbers are equal. If $x_1 = x_2 = \\dots = x_n = x$, and their product is 1, then $x^n = 1$. Since they are positive, $x$ must be 1. In this case, the sum is $n \\times 1 = n$. If the numbers are not all equal, the sum is strictly greater than n. Therefore, the sum is never less than n.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-product-of-n-positive-numbers-is-unity-then-their-sum-is\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"If the product of $n$ positive numbers is unity, then their sum is"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90181","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=90181"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90181\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=90181"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=90181"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=90181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}