{"id":1495,"date":"2020-03-30T05:26:23","date_gmt":"2020-03-30T05:26:23","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1495"},"modified":"2020-04-01T06:35:16","modified_gmt":"2020-04-01T06:35:16","slug":"ibdp-past-year-exam-questions-sequences-and-series","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/sequence-and-series\/ibdp-past-year-exam-questions-sequences-and-series\/","title":{"rendered":"IBDP Past Year Exam Questions &#8211; Sequences and Series"},"content":{"rendered":"<p><strong>1. [M04\/P1]<\/strong><\/p>\r\n<p>A geometric series has a negative common ratio. The sum of the first two terms is 6. The sum to infinity is 8. Find the common ratio and the first term.<\/p>\r\n<div id=\"link1-link-1495\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link1-content-1495\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1496\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-18.jpg\" alt=\"\" width=\"943\" height=\"289\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-18.jpg 943w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-18-300x92.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-18-768x235.jpg 768w\" sizes=\"auto, (max-width: 943px) 100vw, 943px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>2. [M04\/P2]<\/strong><\/p>\r\n<p>The three terms a, 1, b are in arithmetic progression. The three terms 1, a, b are in geometric progression. Find the value of a and of b given that a \u2260 b .<\/p>\r\n<div id=\"link2-link-1495\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link2-content-1495\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1499\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-12.jpg\" alt=\"\" width=\"923\" height=\"122\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-12.jpg 923w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-12-300x40.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-12-768x102.jpg 768w\" sizes=\"auto, (max-width: 923px) 100vw, 923px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>3. [M99\/P1]<\/strong><\/p>\r\n<p>The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12. Find the first term, \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math>\r\n , and the common difference,\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>d<\/mi><\/math>\r\n , of the sequence.<\/p>\r\n<div id=\"link2-link-1495\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link2-content-1495\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1503\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-10.jpg\" alt=\"\" width=\"1061\" height=\"355\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-10.jpg 1061w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-10-300x100.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-10-1024x343.jpg 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-10-768x257.jpg 768w\" sizes=\"auto, (max-width: 1061px) 100vw, 1061px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>4. [M99\/P2]<\/strong><\/p>\r\n<p>The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>6<\/mn><mn>13<\/mn><\/mfrac><\/math>\r\n . If each term of this sequence is positive, and the product of the first term and the third term is 32, find the sum of the first 100 terms of this sequence.\u00a0 \u00a0[7 marks]<\/p>\r\n<div id=\"link4-link-1495\" class=\"sh-link link4-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link4', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link4-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link4-content-1495\" class=\"sh-content link4-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1505\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-11.jpg\" alt=\"\" width=\"1064\" height=\"521\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-11.jpg 1064w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-11-300x147.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-11-1024x501.jpg 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-11-768x376.jpg 768w\" sizes=\"auto, (max-width: 1064px) 100vw, 1064px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>5. [M98\/P1]<\/strong><\/p>\r\n<p>The first, second and the\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math>\r\n the terms of an arithmetic sequence are 2 , 6 , and 58 , respectively.<\/p>\r\n<p>(a) Find the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math>\r\n .<\/p>\r\n<p>(b) For that value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math>\r\n , find the exact value of the sum of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math>\r\n terms of a geometric sequence whose first term is 2 and common ratio is\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/math>\r\n .<\/p>\r\n<div id=\"link5-link-1495\" class=\"sh-link link5-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link5', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link5-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link5-content-1495\" class=\"sh-content link5-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>6. [M08\/P1\/TZ1]<\/strong><\/p>\r\n<p>A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence. The angle of the largest sector is twice the angle of the smallest sector.<br \/>\r\nFind the size of the angle of the smallest sector.<\/p>\r\n<div id=\"link6-link-1495\" class=\"sh-link link6-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link6', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link6-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link6-content-1495\" class=\"sh-content link6-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1508\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-12.jpg\" alt=\"\" width=\"932\" height=\"1054\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-12.jpg 932w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-12-265x300.jpg 265w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-12-905x1024.jpg 905w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-12-768x869.jpg 768w\" sizes=\"auto, (max-width: 932px) 100vw, 932px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>7. [N08\/P1\/TZ0]<\/strong><\/p>\r\n<p>An 81 metre rope is cut into n pieces of increasing lengths that form an arithmetic sequence with a common difference of d metres. Given that the lengths of the shortest and longest pieces are 1.5 metres and 7.5 metres respectively, find the values of n and d .<\/p>\r\n<div id=\"link7-link-1495\" class=\"sh-link link7-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link7', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link7-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link7-content-1495\" class=\"sh-content link7-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1510\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-8.jpg\" alt=\"\" width=\"928\" height=\"194\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-8.jpg 928w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-8-300x63.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-8-768x161.jpg 768w\" sizes=\"auto, (max-width: 928px) 100vw, 928px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>8. [N08\/P2\/TZ0]<\/strong><\/p>\r\n<p>A geometric sequence has a first term of 2 and a common ratio of 1.05. Find the value of the smallest term which is greater than 500.<\/p>\r\n<div id=\"link8-link-1495\" class=\"sh-link link8-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link8', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link8-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link8-content-1495\" class=\"sh-content link8-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1512\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-10.jpg\" alt=\"\" width=\"1006\" height=\"240\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-10.jpg 1006w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-10-300x72.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-10-768x183.jpg 768w\" sizes=\"auto, (max-width: 1006px) 100vw, 1006px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>9. [N12\/P2\/TZ0]<\/strong><\/p>\r\n<p>Find the sum of all the multiples of 3 between 100 and 500.\u00a0 \u00a0[4 marks]<\/p>\r\n<div id=\"link9-link-1495\" class=\"sh-link link9-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link9', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link9-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link9-content-1495\" class=\"sh-content link9-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1514\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-11.jpg\" alt=\"\" width=\"934\" height=\"1207\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-11.jpg 934w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-11-232x300.jpg 232w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-11-792x1024.jpg 792w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-11-768x992.jpg 768w\" sizes=\"auto, (max-width: 934px) 100vw, 934px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>10. [N15\/P1\/TZ1]<\/strong><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1516\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-6.jpg\" alt=\"\" width=\"1126\" height=\"1101\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-6.jpg 1126w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-6-300x293.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-6-1024x1001.jpg 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-6-768x751.jpg 768w\" sizes=\"auto, (max-width: 1126px) 100vw, 1126px\" \/><\/p>\r\n<div id=\"link10-link-1495\" class=\"sh-link link10-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link10', 1495, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link10-toggle-1495\">Solution<\/span><\/a><\/div><div id=\"link10-content-1495\" class=\"sh-content link10-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1517\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12S-1-scaled.jpg\" alt=\"\" width=\"991\" height=\"2560\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12S-1-scaled.jpg 991w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12S-1-116x300.jpg 116w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12S-1-396x1024.jpg 396w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12S-1-768x1984.jpg 768w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12S-1-595x1536.jpg 595w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12S-1-793x2048.jpg 793w\" sizes=\"auto, (max-width: 991px) 100vw, 991px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>11. [M95\/P2]<\/strong><\/p>\r\n<p>(a) The fifth, seventh and twelfth terms of the arithmetic sequence\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>&#160;<\/mo><mo>,<\/mo><mo>&#160;<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>&#160;<\/mo><mo>,<\/mo><mo>&#160;<\/mo><msub><mi>a<\/mi><mrow><mn>3<\/mn><mo>&#160;<\/mo><mo>,<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><\/mrow><\/msub><\/math>\r\n are consecutive terms of a geometric sequence. Find the common ratio of the geometric sequence.\u00a0 \u00a0[9 marks]<\/p>\r\n<p>(b) The sum of first\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math>\r\n positive integers can be written as<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mo>+<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>+<\/mo><mi>k<\/mi><mo>=<\/mo><mfrac><mrow><mi>k<\/mi><mfenced><mrow><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfenced><\/mrow><mn>2<\/mn><\/mfrac><\/math>\r\n.<\/p>\r\n<p>Given\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>&#8712;<\/mo><mi mathvariant=\"normal\">&#8469;<\/mi><\/math>\r\n find, in terms of\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math>\r\n , the sum of the integers between 1 and\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>15<\/mn><mi>n<\/mi><\/math>\r\n inclusive which are not divisible by 3 or 5. Simplify your answer as much as possible.\u00a0 \u00a0[9 marks]<\/p><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"1. [M04\/P1] A geometric series has a negative common ratio. The sum of the first two terms is 6. The sum to infinity is 8. Find the common ratio and the first term. 2. [M04\/P2] The three terms a, 1, b are in arithmetic progression. The three terms 1, a, b are in geometric progression. [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1495","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-sequence-and-series","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1495","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1495"}],"version-history":[{"count":12,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1495\/revisions"}],"predecessor-version":[{"id":1519,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1495\/revisions\/1519"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1495"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}