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It explains the domain of a sequence using natural numbers (positive integers) and maps domain elements to corresponding sequence terms. The chapter also discusses the sum of finite sequences and the idea of a series for infinite sums, then presents arithmetic sequences and derives the general nth-term formula using a common difference d.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-10-sequences-and-series-definition-of-a-sequence-and-its-domain/605558/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/chapter-10-sequences-and-series-definition-of-a-sequence-and-its-domain/605558.png","ImageObject",300,407,{"name":92,"@type":93},"WPS_1790064749","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-23","2026-10-16",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the domain of a sequence, and how does it relate to terms like a1 and a2?","Question",{"text":112,"@type":113},"The domain of a sequence is the natural numbers (positive integers). The number 1 in the domain corresponds to the first term a1, 2 corresponds to a2, and so on, so the nth term corresponds to the nth natural number.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do you define the sum of a finite sequence versus an infinite sequence?",{"text":117,"@type":113},"The sum of a sequence is the total of its terms. If the sequence stops at a particular term (finite), it is called a finite sum; if the sequence is infinite, the sum of that infinite sequence is called a series.",{"name":119,"@type":110,"acceptedAnswer":120},"What formula represents an arithmetic sequence’s nth term?",{"text":121,"@type":113},"If an arithmetic sequence has first term a1 and common difference d, the nth term is an = a1 + (n − 1)d. Using the example where the terms increase by 7, this becomes an = 3 + (n − 1)7.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},605558,1792482616,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},3985747859154,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Chapter 10  \nSequences and Series  \nDefinition of a Sequence and its Domain, the Sum of a Sequence  \nLong ago in a galaxy far, far away, (well, Germany is probably a long way away from where you are, and 1787 is probably a long time ago to most of us), there was a young boy named Carl Friedrich Gauss who was, along with his classmates, driving their teacher to his wits’ end. The story goes that the teacher, in order to quiet the class and try to maintain order, gave the class the assignment to add all the numbers from 1 to 100. He figured that even the brightest would probably take an hour to get the answer, and he could have some peace and quiet.  \nTo his surprise and dismay, young Gauss marched up to the front and handed in his answer within half a minute, and the answer was correct. What Gauss had done was arrange the numbers from 1 to 100 in a sequential order and paired to largest number with the smallest (100 and 1), then the next largest number and the next smallest (99 and 2), until all the numbers in the sequence had been paired with another.  \nGauss realized that he had 50 pairs of numbers, and that each pair added to 101. Simply multiplying  \n50 × 101 = 5050,  \nhe had his answer. I’m sure the rest of the classmates cheered, and the teacher realized that what he thought was a punitive assignment had backfired.  \nWhile I’m sure the study of sequences of numbers happened long before Gauss, the story shows how rapid computation of sums was still fairly unknown at his time.  \nLet’s start with defining a sequence, and focus on what the domain of a sequential function is.  \nDefinition: A sequence of numbers has a first term and succeeding terms, and we call these  \na1, a2, a3, and so on.  \nIf the sequence is finite (that is, that it stops somewhere) we call this stopping point the nth term, and we say that the number of terms in the sequence is that particular n. If the sequence is infinite, there is no specific nth term, and we denote this infinite sequence by a1, a2, a3,…, that is, we put the “dot, dot, dot” at the end of the list.  \nThe domain of a sequence is what we call the “natural numbers”, otherwise known as the positive integers. The 1 of the domain corresponds to the first term, or a1 ; the 2 of the domain corresponds to the second term, or a2, and so on. Then of the domain, some natural number, corresponds to the nth term.  \nHere are some examples of sequences:  \n2 5 17 8 -24 2    \n(Notice in this one that there is no apparent pattern, and a sequence doesn’t need one). Here are some with patterns:  \n2  \n1  \n4  \n−3  \n6  \n5  \n8  \n10  \n−7  \n12   9    \nAgain, notice the three dots after each of these sequences? Those dots (“dot dot dot”) indicate that the sequence goes on indefinitely, that it never comes to an end.  \nWe can see a pattern with the first of these two sequences, in that we go from one even number in the positive integers to the next. We “skip” the odd numbers. It’s easy to see that after the 12 comes 14, and then 16, and so on. Again, the“. . .”means that we never stop with this sequence; it has no finite end.  \nIn the third sequence, we are using odd numbers, and jumping from positive to negative and back to positive again, and so on. Again, we can go on forever because of the “. . .”(dot dot dot).  \nThe sum of a sequence is simply adding the sequence numbers together.  \nIf our sequence is  \n1 2 3 4 5 (and stops at 5), then the sum of this sequence is simply  \n1 + 2 + 3 + 4 + 5 = 15.  \nWe call this a finite sum, and denote it as S 5 .  \nIf the sequence is infinite, then we call the sum of that infinite sequence a series.  \nArithmetic Sequences  \nI like to start off with an example:  \na1 = 3  \na2 = 10  \na3 = 17  \na4 = 24  \n.  \n.  \n.  \nDo you see that this sequence starts at 3, and adds 7 to each succeeding term.? Rather than just jump into a formula for the nth term, I’d like you to try to figure this out more or less on your own. Let’s write the terms again, but this time add the 7 each time:  \na1 = ","cbCait0NO3q9GaO1","https://ap.wps.com/l/cbCait0NO3q9GaO1","pdf",308682,18,"English","# Chapter 10: Sequences and Series\n## Definition of a Sequence and its Domain, the Sum of a Sequence\n## Arithmetic Sequences","[{\"question\":\"What is the domain of a sequence, and how does it relate to terms like a1 and a2?\",\"answer\":\"The domain of a sequence is the natural numbers (positive integers). The number 1 in the domain corresponds to the first term a1, 2 corresponds to a2, and so on, so the nth term corresponds to the nth natural number.\"},{\"question\":\"How do you define the sum of a finite sequence versus an infinite sequence?\",\"answer\":\"The sum of a sequence is the total of its terms. If the sequence stops at a particular term (finite), it is called a finite sum; if the sequence is infinite, the sum of that infinite sequence is called a series.\"},{\"question\":\"What formula represents an arithmetic sequence’s nth term?\",\"answer\":\"If an arithmetic sequence has first term a1 and common difference d, the nth term is an = a1 + (n − 1)d. Using the example where the terms increase by 7, this becomes an = 3 + (n − 1)7.\"}]","Chapter 10 - Sequences and Series - Definition of a Sequence and its Domain | PDF",1792132642,45]