{"id":34585,"date":"2025-06-23T16:58:05","date_gmt":"2025-06-23T16:58:05","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=34585"},"modified":"2025-06-23T16:58:07","modified_gmt":"2025-06-23T16:58:07","slug":"the-sequence","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-sequence\/","title":{"rendered":"The sequence"},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">The sequence<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"354\" height=\"399\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-261.png\" alt=\"\" class=\"wp-image-34586\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-261.png 354w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-261-266x300.png 266w\" sizes=\"auto, (max-width: 354px) 100vw, 354px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The correct answer is: <strong>e. neither increasing nor decreasing<\/strong><\/p>\n\n\n\n<p>To understand why, we examine the behavior of the sequence defined by<\/p>\n\n\n\n<p>{1n\u2212(\u22121)n}n=0\u221e\\left\\{ \\frac{1}{n &#8211; (-1)^n} \\right\\}_{n=0}^{\\infty}<\/p>\n\n\n\n<p>This sequence depends on the parity of nn, because (\u22121)n(-1)^n alternates between 1 and -1 depending on whether nn is even or odd.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For even nn, (\u22121)n=1(-1)^n = 1, so the general term becomes 1n\u22121\\frac{1}{n &#8211; 1}<\/li>\n\n\n\n<li>For odd nn, (\u22121)n=\u22121(-1)^n = -1, making the term 1n+1\\frac{1}{n + 1}<\/li>\n<\/ul>\n\n\n\n<p>This split means the sequence alternates between two sub-sequences: one involving 1n\u22121\\frac{1}{n &#8211; 1} and the other involving 1n+1\\frac{1}{n + 1}. Let&#8217;s compute the first several terms:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>n=0n = 0: 10\u22121=\u22121\\frac{1}{0 &#8211; 1} = -1<\/li>\n\n\n\n<li>n=1n = 1: 11+1=12\\frac{1}{1 + 1} = \\frac{1}{2}<\/li>\n\n\n\n<li>n=2n = 2: 12\u22121=1\\frac{1}{2 &#8211; 1} = 1<\/li>\n\n\n\n<li>n=3n = 3: 13+1=14\\frac{1}{3 + 1} = \\frac{1}{4}<\/li>\n\n\n\n<li>n=4n = 4: 14\u22121=13\\frac{1}{4 &#8211; 1} = \\frac{1}{3}<\/li>\n\n\n\n<li>n=5n = 5: 15+1=16\\frac{1}{5 + 1} = \\frac{1}{6}<\/li>\n<\/ul>\n\n\n\n<p>Now notice how the values rise and fall inconsistently. From n=0n = 0 to n=1n = 1, the value jumps from -1 to 0.5. Then it goes to 1, back to 0.25, up to 0.333, and down to 0.167. These fluctuations show neither a strictly increasing nor decreasing pattern. Furthermore, even considering magnitude, the values do not follow a consistent trend in any direction.<\/p>\n\n\n\n<p>Therefore, the sequence fails all definitions for monotonic behavior, making option <strong>e<\/strong> the only valid choice.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"722\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-288.jpeg\" alt=\"\" class=\"wp-image-34587\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-288.jpeg 722w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-288-212x300.jpeg 212w\" sizes=\"auto, (max-width: 722px) 100vw, 722px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The sequence The Correct Answer and Explanation is: The correct answer is: e. neither increasing nor decreasing To understand why, we examine the behavior of the sequence defined by {1n\u2212(\u22121)n}n=0\u221e\\left\\{ \\frac{1}{n &#8211; (-1)^n} \\right\\}_{n=0}^{\\infty} This sequence depends on the parity of nn, because (\u22121)n(-1)^n alternates between 1 and -1 depending on whether nn is even [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-34585","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34585","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=34585"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34585\/revisions"}],"predecessor-version":[{"id":34588,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34585\/revisions\/34588"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=34585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=34585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=34585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}