{"id":218126,"date":"2025-05-23T08:54:14","date_gmt":"2025-05-23T08:54:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=218126"},"modified":"2025-05-23T08:54:17","modified_gmt":"2025-05-23T08:54:17","slug":"which-expression-is-equivalent-to-the-given-expression-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/23\/which-expression-is-equivalent-to-the-given-expression-3\/","title":{"rendered":"Which expression is equivalent to the given expression"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Which expression is equivalent to the given expression? 6 (2i \u2013 3) i=-3<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">YOUR ANSWER CORRECT ANSWER<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">5 (2i \u2013 3) i=-2<br>5 9+ (2\u20133) i=-2<br>9+ (2i \u2013 3) i=-2<br>5<br>5 -9+ (2i \u2013 3) ) i=-3<\/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<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>6 (2i \u2013 3) with i = -3<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But this is ambiguous because it looks like summation notation or evaluation of an expression at a specific value of $i$. I assume the original question is about a summation or series:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Is the original expression:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\sum_{i=-3}^6 (2i &#8211; 3)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">or something similar?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Your provided expression is unclear, so I&#8217;ll interpret it as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Find an expression equivalent to:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\sum_{i=-3}^6 (2i &#8211; 3)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And your answer choices look like variants of sums with different indices or terms.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Understand the sum notation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\sum_{i=a}^b f(i)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">means sum $f(i)$ as $i$ goes from $a$ to $b$, including both $a$ and $b$.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the sum $\\sum_{i=-3}^6 (2i &#8211; 3)$<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We want to find the sum of terms:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>(2(-3) &#8211; 3) + (2(-2) &#8211; 3) + \\cdots + (2(6) &#8211; 3)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First, find the number of terms:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From $-3$ to $6$, inclusive, there are $6 &#8211; (-3) + 1 = 6 + 3 + 1 = 10$ terms.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Use formula for sum of linear sequence<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Rewrite the sum:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>S = \\sum_{i=-3}^6 (2i &#8211; 3) = 2 \\sum_{i=-3}^6 i &#8211; 3 \\sum_{i=-3}^6 1<br>$$<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\sum_{i=-3}^6 i$ is the sum of integers from $-3$ to $6$.<\/li>\n\n\n\n<li>$\\sum_{i=-3}^6 1$ counts the number of terms.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Number of terms: 10 (as above).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate $\\sum_{i=-3}^6 i$:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sum of integers from $m$ to $n$ is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{(n + m)(n &#8211; m + 1)}{2}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Plug in $m = -3$, $n = 6$:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{(6 + (-3))(6 &#8211; (-3) + 1)}{2} = \\frac{(3)(6 + 3 + 1)}{2} = \\frac{3 \\times 10}{2} = \\frac{30}{2} = 15<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>S = 2 \\times 15 &#8211; 3 \\times 10 = 30 &#8211; 30 = 0<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Compare to answer choices<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Your answer choices are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$5 \\sum_{i=-2} (2i &#8211; 3)$<\/li>\n\n\n\n<li>$5 \\cdot 9 + \\sum_{i=-2} (2 &#8211; 3)$<\/li>\n\n\n\n<li>$\\sum_{i=-2} (2i &#8211; 3)$<\/li>\n\n\n\n<li>$5$<\/li>\n\n\n\n<li>$5 &#8211; 9 + \\sum_{i=-3} (2i &#8211; 3)$<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">Since the sum from $-3$ to 6 equals 0, let&#8217;s test which of these might equal 0.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Option 3: $\\sum_{i=-2} (2i &#8211; 3)$ \u2014 incomplete notation (no upper limit). Assuming it&#8217;s $\\sum_{i=-2}^5 (2i &#8211; 3)$ or similar, this is different from the original sum and likely not equal to 0.<\/li>\n\n\n\n<li>Option 4: $5$ \u2014 this is just a number 5, which is not equal to 0.<\/li>\n\n\n\n<li>Option 5: $5 &#8211; 9 + \\sum_{i=-3} (2i &#8211; 3)$ \u2014 again incomplete summation notation, but $5 &#8211; 9 = -4$, so this would depend on the sum.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Without clearer options, the only exact equivalent sum expression would be:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\sum_{i=-3}^6 (2i &#8211; 3) = 0<br>$$<\/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<ul class=\"wp-block-list\">\n<li>The sum $\\sum_{i=-3}^6 (2i &#8211; 3)$ is equivalent to 0.<\/li>\n\n\n\n<li>This is found by separating the sum into two parts and calculating each with formulas for arithmetic sums.<\/li>\n\n\n\n<li>The answer choices provided are incomplete or unclear, but none seems to equal zero directly.<\/li>\n\n\n\n<li>The key is that summing a linear expression over a symmetric range around zero or near it can lead to zero because positive and negative terms cancel out.<\/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\">Detailed Explanation (300 words)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The original expression $\\sum_{i=-3}^6 (2i &#8211; 3)$ involves summing a linear function of $i$, specifically $2i &#8211; 3$, as $i$ goes from $-3$ to 6 inclusive. To find an equivalent expression or evaluate it, you can break the summation into simpler parts using properties of summations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First, the summation can be split into the sum of two parts:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\sum_{i=-3}^6 (2i &#8211; 3) = \\sum_{i=-3}^6 2i &#8211; \\sum_{i=-3}^6 3<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Because summation is linear, constants can be factored out:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>= 2 \\sum_{i=-3}^6 i &#8211; 3 \\sum_{i=-3}^6 1<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Next, evaluate each summation separately:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$\\sum_{i=-3}^6 i$ is the sum of integers from $-3$ up to $6$. Using the formula for the sum of an arithmetic sequence, which is $\\frac{(n + m)(n &#8211; m + 1)}{2}$ where $m = -3$ and $n = 6$, you find the sum is 15.<\/li>\n\n\n\n<li>$\\sum_{i=-3}^6 1$ simply counts the number of terms. From $-3$ to 6, there are 10 terms.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Plugging these values back in:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>2 \\times 15 &#8211; 3 \\times 10 = 30 &#8211; 30 = 0<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So the sum evaluates to zero.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This explains that the sum of a linear function over this symmetric range cancels out exactly.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Regarding the answer choices, without exact summation bounds or clarifications, none directly matches this calculation except the expression that fully captures the sum from $-3$ to $6$. Simplifying or shifting indices as in other choices will change the value, so the correct answer is the original summation $\\sum_{i=-3}^6 (2i &#8211; 3)$, which equals zero.<\/p>\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-152.png\" alt=\"\" class=\"wp-image-218127\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Which expression is equivalent to the given expression? 6 (2i \u2013 3) i=-3 YOUR ANSWER CORRECT ANSWER 5 (2i \u2013 3) i=-25 9+ (2\u20133) i=-29+ (2i \u2013 3) i=-255 -9+ (2i \u2013 3) ) i=-3 The Correct Answer and Explanation 6 (2i \u2013 3) with i = -3 But this is ambiguous because it looks [&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-218126","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218126","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=218126"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218126\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=218126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=218126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=218126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}