[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-573393-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-573393-en":126},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","theorem-booklet-grade-1112-mathematics-revision-guide","Theorem-Booklet - Grade 11/12 Mathematics Revision Guide","","Mathematics revision guide compiled for GA Mac Tavish (KZN Pinetown District), supporting Grade 11 and Grade 12 learners. Covers important definitions, proofs, theorems and formulae for Paper 1 and Paper 2, aligned with CAPS. Includes proofs for sums of arithmetic and geometric sequences, plus an information sheet of key formulae such as quadratic, interest, depreciation, series and probability identities. Also lists Paper 2 geometry, trigonometry and statistics formula summaries including line, circle, sine/cosine rules, and variance/standard deviation calculations.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/paper-templates/","Paper Templates",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/theorem-booklet-grade-1112-mathematics-revision-guide/573393/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/theorem-booklet-grade-1112-mathematics-revision-guide/573393.png","ImageObject",442,249,{"name":42,"@type":43},"Jacob","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-15","2026-10-14",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What is included in the revision guide for Paper 1 and Paper 2?","Question",{"text":62,"@type":63},"It includes important mathematical definitions, proofs, theorems and formulae for both Paper 1 and Paper 2, suitable for Grade 11 and Grade 12 learners.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"Does the guide contain proofs for sequence-sum formulae?",{"text":67,"@type":63},"Yes. It provides Proof 1 for the sum of an arithmetic sequence and Proof 2 for the sum to n terms of a geometric sequence.",{"name":69,"@type":60,"acceptedAnswer":70},"Which Paper 2 topics have formula summaries?",{"text":71,"@type":63},"Paper 2 summaries include formulas for straight lines (various line forms and gradients), circles, trigonometry rules such as sine and cosine rules, and statistics items like variance and standard deviation calculations.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},573393,1792053095,{"code":4,"msg":81,"data":82},"success",[83,88,93,98,103,108,113,118,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},11,"Presentations",90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social Media",50,"social-media",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},17,"Forms",40,"forms",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},18,"Letters",30,"letters",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":120,"slug":121},21,5,"papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":119,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":22,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":135,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":141},962084931830,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Compiled by: GA Mac Tavish KZN – Pinetown District Grade 11/12 (2019)  \nM Mpisi  \nKwaZulu-Natal  \nPINETOWN DISTRICT  \nThis revision guide contains important mathematical definitions, proofs, theorems and formula for  \nPaper 1 and Paper 2 .  \nThis content is CAPS compliant and suitable for Grade 11 and Grade 12 learners.  \nPage 1 of 19  \nEducators may use, modify, upload, download, and share this content, but you must acknowledge Pinetown District, the author and contributors.  \nCompiled by: GA Mac Tavish KZN – Pinetown District Grade 11/12 (2019)  \nM Mpisi  \nPAPER 1  \nArithmetic and geometric series  \nProof 1: Prove that the sum of terms of an arithmetic sequence is given by the formula:  \nn  \nSn 􀀠 [2a 􀀎 (n 􀀐1)d ]  \n2  \nSn 􀀠 a 􀀎 (a 􀀎 d) 􀀎 ....a. 􀀎 (n 􀀐1)d  \nSn 􀀠 [a 􀀎 (n 􀀐1)d] 􀀎 [a 􀀎 (n 􀀐 2)d ]􀀎 ....􀀎 a  \n2Sn 􀀠 2a 􀀎 (n 􀀐1)d 􀀎 ....2. a 􀀎 (n 􀀐1)d 2Sn 􀀠 n[2a 􀀎 (n 􀀐1)d]  \nn  \nSn 􀀠 2 [2a 􀀎 (n 􀀐1)d ]  \nProve 1 (alternative):  \nThe general term (last term) of an arithmetic series is Tn = a + (n – 1)d = l Sn = a + [a + d] + [a + 2d] + …+ [l – d] + l …equation 1  \nIn reverse: Sn = [a + (n – 1)d] + [a + (n – 2)d] + [a + (n – 3)d] + …+ [a + d] + a  \nSn = l + [l – d] + [l – 2d] + . . . + [a + d] + a …equation 2 Adding equation 1 and equation 2:  \n2Sn = [a + l] + [a + l] +   + [a + l] to n terms  \n2Sn = n[a + l]  \nn  \n∴Sn = [a 􀀎 l]  \n2  \nn  \nReplacing l with a + (n – 1)d: ∴Sn = [2a 􀀎 (n 􀀐1)d]  \n2  \nProof 2: Prove that the sum to n terms ofa geometric sequence is given by:  \na (rn 􀀐1)  \nSn 􀀠 ; r 􀁺 1  \nr 􀀐1  \nSn 􀀠 a 􀀎 ar 􀀎 ar 2 􀀎 ......... 􀀎 arn􀀐1  \nrSn 􀀠 ar 􀀎 ar 2 􀀎 ......... 􀀎 arn􀀐1 􀀎 arn  \nSn 􀀐 rSn 􀀠 a 􀀐 arn Sn (1􀀐 r) 􀀠 a(1􀀐 rn )  \nS 􀀠 a(1􀀐 rn ) n 1 􀀐 r  \nPage 2 of 19  \nEducators may use, modify, upload, download, and share this content, but you must acknowledge Pinetown District, the author and contributors.  \n\n| Compiled by: GA Mac Tavish KZN – Pinetown District Grade 11/12 (2019)\u003Cbr>M Mpisi\u003Cbr>Information Sheet – formula for paper 1 |  |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| x 􀀠 | 􀀐b 􀁲 b2 􀀐 4ac\u003Cbr>2a |  | \u003Cbr>Quadratic Formula. Used to find roots of a quadratic (parabola) . Parts of this formula can be used to find x-value ofTP. Also the discriminant (Δ = b2 – 4ac) for Nature of Roots. |  |  |  |  |  |\n| A 􀀠 P(1 􀀎 ni)\u003Cbr>\u003Cbr>Simple Interest / growth.\u003Cbr>Tn 􀀠 a 􀀎 (n 􀀐1)d\u003Cbr>\u003Cbr>Linear patterns.\u003Cbr>Arithmetic Progression. Finding the general term of an AP.\u003Cbr>*Common difference. |  |  | A 􀀠 P(1 􀀐 ni)\u003Cbr>\u003Cbr>Simple Depreciation / straight line depreciation.\u003Cbr>A 􀀠 P(1 􀀐 i)n\u003Cbr>\u003Cbr>Reducing / diminished balance depreciation.\u003Cbr>S 􀀠 n {2a 􀀎 (n 􀀐1)d} n 2\u003Cbr>\u003Cbr>Linear patterns. Arithmetic Series. Finding the sum ofan Arithmetic Series. LEARN THE PROOF FOR THIS. |  |  |  | A 􀀠 P(1 􀀎 i)n\u003Cbr>\u003Cbr>Compound Interest / growth. Also used for inflation. |  |\n| Tn 􀀠 arn􀀐1\u003Cbr>\u003Cbr>Geometric patterns. Geometric Progression. Finding the general term of a GP.\u003Cbr>*Common ratio. |  |  | Sn 􀀠 a~~ ~~􀀋rr~~ ~~1􀀌 ; r 􀁺 1 |  | | S 􀀠  a 􀁦 1 􀀐 r\u003Cbr>\u003Cbr>Sum to infinity |  | ; 􀀐1􀀟 r 􀀟1\u003Cbr>\u003Cbr>Condition for\u003Cbr>convergence. |\n|  |  |  | Geometric patterns. Geometric Series. Finding the sum ofa GS. LEARN THE PROOF FOR THIS. |  |  |  |  |  |\n| x 􀀋1 􀀎 i 􀀌n 􀀐1 F 􀀠 ~~ ~~\u003Cbr>i \u003Cbr>Future Value Annuity. Investments and savings. Used when working with sinking funds.\u003Cbr>f '(x) 􀀠 lim f (x 􀀎 h) 􀀐 f (x)\u003Cbr>h􀁯0 h |  |  |  | P 􀀠 x 1 􀀐 (1 􀀎 i)􀀐n i \u003Cbr>Present Value Annuity. Loans. Also used when living off an inheritance / large amount of money.\u003Cbr>|  |  |  |  |\n| P(A) 􀀠 nn(􀀋AS􀀌) |  | |  | P (A orB) = P(A) + P(B)–P(A andB)\u003Cbr>\u003Cbr>Fundamental Probability Identity. |  |  |  |  |\n| Probability definition. |  |  |  |  |  |  |  |  |\n| Page 3 of 19\u003Cbr>Educators may use, modify, upload, download, and share this content, but you must acknowledge Pinetown District, the author and contributors. |  |  |  |  |  |  |  |  |\n\n\n| Compiled by: GA Mac Tavish KZN – Pinetown District Grade 11/12 (2019)\u003Cbr>M Mpisi\u003Cbr>Information Sheet – formula for paper 2 |  |  |  ","cbCainvGvG3DgESH","https://ap.wps.com/l/cbCainvGvG3DgESH","pdf",1237996,19,"English","# Paper 1\n## Arithmetic and geometric series\n# Paper 2\n## Straight lines and circles\n## Trigonometry rules\n## Statistics and probability","[{\"question\":\"What is included in the revision guide for Paper 1 and Paper 2?\",\"answer\":\"It includes important mathematical definitions, proofs, theorems and formulae for both Paper 1 and Paper 2, suitable for Grade 11 and Grade 12 learners.\"},{\"question\":\"Does the guide contain proofs for sequence-sum formulae?\",\"answer\":\"Yes. It provides Proof 1 for the sum of an arithmetic sequence and Proof 2 for the sum to n terms of a geometric sequence.\"},{\"question\":\"Which Paper 2 topics have formula summaries?\",\"answer\":\"Paper 2 summaries include formulas for straight lines (various line forms and gradients), circles, trigonometry rules such as sine and cosine rules, and statistics items like variance and standard deviation calculations.\"}]","Theorem-Booklet - Grade 11/12 Mathematics Revision Guide | PDF",1791938840,7]