{"id":217906,"date":"2025-05-22T17:00:01","date_gmt":"2025-05-22T17:00:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217906"},"modified":"2025-05-22T19:30:28","modified_gmt":"2025-05-22T19:30:28","slug":"h-o-t-problems-use-higher-order-thinking-skills-32","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/h-o-t-problems-use-higher-order-thinking-skills-32\/","title":{"rendered":"Create an arithmetic sequence with a common difference of -10. 33."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">H.O.T. Problems Use Higher-Order Thinking Skills 32. OPEN ENDED Create an arithmetic sequence with a common difference of -10. 33. CCSS PERSEVERANCE Find the value of x that makes x + 8, 4x + 6, and 3x the first three terms of an arithmetic sequence. 34. REASONING Compare and contrast the domain and range of the linear functions described by Ax + By = C and a = a + (n &#8211; 1)d. 35. CHALLENGE Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain. If yes, find the common difference and the next three terms. a. 2x + 1, 3x + 1, 4x + 1\u2026 b. 2x, 4x, 8x, \u2026<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">32. Create an arithmetic sequence with a common difference of -10.<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><br>An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the <strong>common difference<\/strong> $d$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the common difference $d = -10$, an example arithmetic sequence could be:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$50, 40, 30, 20, 10, 0, -10, \\dots$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Explanation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Starting from 50 (first term), each subsequent term is found by subtracting 10.<\/li>\n\n\n\n<li>$50 &#8211; 10 = 40$,<\/li>\n\n\n\n<li>$40 &#8211; 10 = 30$, and so forth.<br>Thus, this sequence fits the definition of an arithmetic sequence with a common difference of $-10$.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">33. Find the value of $x$ that makes $x + 8$, $4x + 6$, and $3x$ the first three terms of an arithmetic sequence.<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><br>For these three terms to form an arithmetic sequence, the difference between consecutive terms must be the same.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set the common difference $d$ as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>(4x + 6) &#8211; (x + 8) = 3x &#8211; (4x + 6)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Simplify both sides:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>(4x + 6) &#8211; (x + 8) = 3x + 6 &#8211; 4x = -x + 6<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>(4x + 6) &#8211; (x + 8) = 3x + (6 &#8211; 8) = 3x &#8211; 2<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>3x &#8211; (4x + 6) = 3x &#8211; 4x &#8211; 6 = -x &#8211; 6<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set equal:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>3x &#8211; 2 = -x &#8211; 6<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Add $x$ to both sides:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>3x + x &#8211; 2 = -6<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>4x &#8211; 2 = -6<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Add 2 to both sides:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>4x = -4<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Divide both sides by 4:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>x = -1<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">34. Compare and contrast the domain and range of the linear functions described by $Ax + By = C$ and $a = a + (n &#8211; 1)d$.<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Function $Ax + By = C$:<\/strong><br>This is the standard form of a <strong>linear equation<\/strong> in two variables $x$ and $y$. The graph is a straight line.<\/li>\n\n\n\n<li><strong>Domain:<\/strong> All real values of $x$ for which $y$ can be found. Usually, $x \\in \\mathbb{R}$, unless restricted. So domain is typically all real numbers.<\/li>\n\n\n\n<li><strong>Range:<\/strong> Similarly, since the line extends infinitely in both directions, $y$ can take all real values. So the range is also all real numbers.<\/li>\n\n\n\n<li><strong>Function $a = a + (n &#8211; 1)d$:<\/strong><br>This formula defines the <strong>nth term of an arithmetic sequence<\/strong>.<\/li>\n\n\n\n<li><strong>Domain:<\/strong> The input variable $n$ is the term number, so $n$ is restricted to <strong>positive integers<\/strong> (1, 2, 3, \u2026), since terms in a sequence are counted discretely.<\/li>\n\n\n\n<li><strong>Range:<\/strong> The values of $a$ (the terms of the sequence) depend on $a$ (first term) and $d$ (common difference). The range is the set of these term values and can be infinite or finite depending on $d$.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Contrast:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The first is a continuous linear function with both domain and range as real numbers.<\/li>\n\n\n\n<li>The second is a discrete function (sequence), with domain as positive integers and range as the terms generated by the formula.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">35. Determine whether each sequence is arithmetic. Write yes or no. Explain. If yes, find the common difference and the next three terms.<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>a. $2x + 1, 3x + 1, 4x + 1, \\dots$<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the difference between consecutive terms:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>(3x + 1) &#8211; (2x + 1) = 3x + 1 &#8211; 2x &#8211; 1 = x<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>(4x + 1) &#8211; (3x + 1) = 4x + 1 &#8211; 3x &#8211; 1 = x<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since the difference between consecutive terms is the same ($x$), this <strong>is<\/strong> an arithmetic sequence with common difference $d = x$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The next three terms would be:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>5x + 1, \\quad 6x + 1, \\quad 7x + 1<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>b. $2x, 4x, 8x, \\dots$<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the difference between consecutive terms:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>4x &#8211; 2x = 2x<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>8x &#8211; 4x = 4x<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since the difference is not constant (2x then 4x), this <strong>is not<\/strong> an arithmetic sequence.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Explanation:<br>An arithmetic sequence requires a constant difference between terms. This sequence doubles each time, indicating it&#8217;s geometric, not arithmetic.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Summary:<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Problem<\/th><th>Answer<\/th><th>Explanation Summary<\/th><\/tr><\/thead><tbody><tr><td>32<\/td><td>Sequence example: 50, 40, 30\u2026<\/td><td>Common difference -10<\/td><\/tr><tr><td>33<\/td><td>$x = -1$<\/td><td>Solve difference equality<\/td><\/tr><tr><td>34<\/td><td>Linear eqn: domain &amp; range real; sequence: domain positive integers, range depends on terms<\/td><td>Continuous vs discrete function<\/td><\/tr><tr><td>35a<\/td><td>Yes; common difference $x$; next terms $5x+1, 6x+1, 7x+1$<\/td><td>Constant difference $x$<\/td><\/tr><tr><td>35b<\/td><td>No<\/td><td>Differences vary; not arithmetic<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-102.png\" alt=\"\" class=\"wp-image-217907\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>H.O.T. Problems Use Higher-Order Thinking Skills 32. OPEN ENDED Create an arithmetic sequence with a common difference of -10. 33. CCSS PERSEVERANCE Find the value of x that makes x + 8, 4x + 6, and 3x the first three terms of an arithmetic sequence. 34. REASONING Compare and contrast the domain and range of [&hellip;]<\/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-217906","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217906","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=217906"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217906\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217906"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217906"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217906"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}