{"id":1388,"date":"2026-01-27T18:20:07","date_gmt":"2026-01-27T08:20:07","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1388"},"modified":"2026-01-28T19:48:39","modified_gmt":"2026-01-28T09:48:39","slug":"year12-math-4-3-1-mathematical-induction","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1388","title":{"rendered":"Year12 MATH 4-3-1 Mathematical Induction"},"content":{"rendered":"\n

In QLD Year 12 Specialist Mathematics (Unit 3), Mathematical Induction<\/strong> is a foundational proof technique used to establish the truth of mathematical statements (propositions) for all natural numbers (n<\/mi>\u2208<\/mo>N<\/mi><\/mrow>n is an element of the natural numbers<\/annotation><\/semantics><\/math>). <\/p>\n\n\n\n

While “sample proportions” are a statistical concept often found in Mathematical Methods (Topic 4, Unit 4), in the context of Specialist Unit 3, Mathematical Induction is applied to sequences, series (summations), and divisibility proofs<\/strong>. <\/p>\n\n\n\n

Here is a detailed breakdown of Mathematical Induction based on the Unit 3 QCE syllabus: <\/p>\n\n\n\n

1. The Structure of Proof by Induction <\/strong><\/p>\n\n\n\n

Induction works like a domino effect: if the first domino falls, and any given domino falling knocks over the next, all dominoes will fall. The formal process involves four steps: <\/p>\n\n\n\n

    \n
  1. Base Case (n=1 or \ud835\udc5b=\ud835\udc5b0):<\/strong> Prove the proposition is true for the first integer (usually 1).<\/li>\n\n\n\n
  2. Inductive Hypothesis (\ud835\udc5b=\ud835\udc58):<\/strong> Assume the proposition is true for some arbitrary positive integer k<\/mi>k<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  3. Inductive Step (\ud835\udc5b=\ud835\udc58+1):<\/strong> Using the assumption from step 2, prove the proposition is true for k<\/mi>+<\/mo>1<\/mn><\/mrow>k plus 1<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  4. Conclusion:<\/strong> State that since the base case is true, and the k<\/mi>\u2192<\/mo>k<\/mi>+<\/mo>1<\/mn><\/mrow>k right arrow k plus 1<\/annotation><\/semantics><\/math> step is true, the statement is true for all natural numbers.\u00a0<\/li>\n<\/ol>\n\n\n\n

    2. Applications in Unit 3 Specialist Math <\/strong><\/p>\n\n\n\n

    According to QCAA, Induction is used in Topic 1 (Unit 3) for: <\/p>\n\n\n\n

      \n
    • Sums\/Series:<\/strong> Proving formulae for the sum of sequences (e.g., 1<\/mn>+<\/mo>2<\/mn>+<\/mo>3<\/mn>+<\/mo>.<\/mn>.<\/mn>.<\/mn>+<\/mo>n<\/mi>=<\/mo>n<\/mi>(<\/mo>n<\/mi>+<\/mo>1<\/mn>)<\/mo><\/mrow>2<\/mn><\/mfrac><\/mrow>1 plus 2 plus 3 plus point point point plus n equals the fraction with numerator n open paren n plus 1 close paren and denominator 2 end-fraction<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
    • Divisibility:<\/strong> Proving an expression is divisible by a certain number (e.g., 2<\/mn>2<\/mn>n<\/mi><\/mrow><\/msup>\u2212<\/mo>1<\/mn><\/mrow>2 raised to the 2 n power minus 1<\/annotation><\/semantics><\/math> is divisible by 3).<\/li>\n\n\n\n
    • Partial Sums:<\/strong> Proving sum formulas. <\/li>\n<\/ul>\n\n\n\n

      3. Relation to Sample Proportions (Contextual Misinterpretation) <\/strong><\/p>\n\n\n\n

      Based on search results, “sample proportions” (p<\/mi>\u0302<\/mo><\/mover>p hat<\/annotation><\/semantics><\/math>) are not the main focus of induction in Unit 3. Instead, sample proportions are studied under statistical inference<\/em> in Methods\/Specialist to estimate population parameters using confidence intervals (p<\/mi>\u0302<\/mo><\/mover>\u00b1<\/mo>z<\/mi>p<\/mi>\u0302<\/mo><\/mover>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>\u0302<\/mo><\/mover>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>p hat plus or minus z the square root of the fraction with numerator p hat open paren 1 minus p hat close paren and denominator n end-fraction end-root<\/annotation><\/semantics><\/math>). <\/p>\n\n\n\n

      If “induction” and “proportions” appear together, it may refer to: <\/p>\n\n\n\n

        \n
      • Proving a Formula involving nn\ud835\udc5b:<\/strong> Using induction to prove a formula for a variance or expected value that contains<\/em> an n<\/mi>n<\/annotation><\/semantics><\/math> (sample size) parameter.\u00a0<\/li>\n<\/ul>\n\n\n\n

        4. Key Exam Components <\/p>\n\n\n\n

          \n
        • Technology:<\/strong> Non-CAS calculators are typically permitted for Unit 3 exams.<\/li>\n\n\n\n
        • Technique:<\/strong> The proof must rigorously show the Left Hand Side (LHS) equals the Right Hand Side (RHS) for k<\/mi>+<\/mo>1<\/mn><\/mrow>k plus 1<\/annotation><\/semantics><\/math>.\u00a0<\/li>\n<\/ul>\n\n\n\n

          Note: Unit 3 of Specialist Mathematics (QLD) also covers Vectors in 3D, Complex Numbers (De Moivre’s Theorem), and Functions.<\/em> <\/p>\n\n\n\n

          ***************************************************************************************<\/p>\n\n\n\n

          QLD 12\u5e74\u751f\u5c02\u9580\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c83\uff09\u306b\u304a\u3044\u3066\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306f\u3001\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\uff08n\u2208N, n\u306f\u81ea\u7136\u6570\u306e\u5143\uff09\u306b\u5bfe\u3059\u308b\u6570\u5b66\u7684\u547d\u984c\uff08\u547d\u984c\uff09\u306e\u771f\u507d\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u308b\u57fa\u790e\u7684\u306a\u8a3c\u660e\u6280\u6cd5\u3067\u3059\u3002<\/p>\n\n\n\n

          \u300c\u6a19\u672c\u5272\u5408\u300d\u306f\u6570\u5b66\u7684\u65b9\u6cd5\uff08\u30c8\u30d4\u30c3\u30af4\u3001\u30e6\u30cb\u30c3\u30c84\uff09\u3067\u3088\u304f\u898b\u3089\u308c\u308b\u7d71\u8a08\u7684\u6982\u5ff5\u3067\u3059\u304c\u3001\u5c02\u9580\u30e6\u30cb\u30c3\u30c83\u306e\u6587\u8108\u3067\u306f\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306f\u6570\u5217\u3001\u7d1a\u6570\uff08\u548c\uff09\u3001\u304a\u3088\u3073\u5272\u308a\u5207\u308c\u308b\u304b\u3069\u3046\u304b\u306e\u8a3c\u660e\u306b\u9069\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

          QCE\u30e6\u30cb\u30c3\u30c83\u306e\u30b7\u30e9\u30d0\u30b9\u306b\u57fa\u3065\u3044\u3066\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u8a73\u7d30\u306b\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

            \n
          1. \u5e30\u7d0d\u6cd5\u306b\u3088\u308b\u8a3c\u660e\u306e\u69cb\u9020<\/strong><\/li>\n<\/ol>\n\n\n\n

            \u5e30\u7d0d\u6cd5\u306f\u30c9\u30df\u30ce\u5012\u3057\u306e\u3088\u3046\u306b\u6a5f\u80fd\u3057\u307e\u3059\u3002\u6700\u521d\u306e\u30c9\u30df\u30ce\u304c\u5012\u308c\u3001\u5012\u308c\u305f\u30c9\u30df\u30ce\u304c\u6b21\u306e\u30c9\u30df\u30ce\u3092\u5012\u3059\u3068\u3001\u3059\u3079\u3066\u306e\u30c9\u30df\u30ce\u304c\u5012\u308c\u307e\u3059\u3002\u6b63\u5f0f\u306a\u624b\u9806\u306f4\u3064\u306e\u30b9\u30c6\u30c3\u30d7\u304b\u3089\u6210\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

              \n
            1. \u57fa\u672c\u30b1\u30fc\u30b9\uff08n=1 \u307e\u305f\u306f n=n0\uff09\uff1a\u547d\u984c\u304c\u6700\u521d\u306e\u6574\u6570\uff08\u901a\u5e38\u306f 1\uff09\u306b\u5bfe\u3057\u3066\u771f\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e<\/li>\n\n\n\n
            2. \u5e30\u7d0d\u7684\u4eee\u8aac\uff08n=n\uff09\uff1a\u547d\u984c\u304c\u4efb\u610f\u306e\u6b63\u306e\u6574\u6570 kkn \u306b\u5bfe\u3057\u3066\u771f\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002<\/li>\n<\/ol>\n\n\n\n

              3. \u5e30\u7d0d\u7684\u30b9\u30c6\u30c3\u30d7\uff08n=n+1\uff09\uff1a\u30b9\u30c6\u30c3\u30d7 2 \u306e\u4eee\u5b9a\u3092\u7528\u3044\u3066\u3001\u547d\u984c\u304c k+1 \u306b \u5bfe\u3057\u3066\u771f\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

              4. \u7d50\u8ad6\uff1a\u57fa\u672c\u30b1\u30fc\u30b9\u304c\u771f\u3067\u3042\u308a\u3001k\u2192k+1 \u30b9\u30c6\u30c3\u30d7\u304c\u771f\u3067\u3042\u308b\u305f\u3081\u3001\u3053\u306e\u547d\u984c\u306f\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\u306b\u5bfe\u3057\u3066\u771f\u3067\u3042\u308b\u3053\u3068\u3092\u8ff0\u3079\u307e\u3059\u3002<\/p>\n\n\n\n

                \n
              1. \u30e6\u30cb\u30c3\u30c83 \u5c02\u9580\u6570\u5b66\u306b\u304a\u3051\u308b\u5fdc\u7528<\/strong><\/li>\n<\/ol>\n\n\n\n

                QCAA\u306b\u3088\u308b\u3068\u3001\u30c8\u30d4\u30c3\u30af1\uff08\u30e6\u30cb\u30c3\u30c83\uff09\u3067\u306f\u3001\u5e30\u7d0d\u6cd5\u306f\u4ee5\u4e0b\u306e\u76ee\u7684\u3067\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

                \u548c\uff0f\u7d1a\u6570\uff1a\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u306e\u8a3c\u660e\uff08\u4f8b\uff1a1<\/mn>+<\/mo>2<\/mn>+<\/mo>3<\/mn>+<\/mo>.<\/mi>.<\/mi>.<\/mi>+<\/mo>n<\/mi>=<\/mo>n<\/mi>(<\/mo>n<\/mi>+<\/mo>1<\/mn>)<\/mo><\/mrow>2<\/mn><\/mfrac>)<\/mo><\/mrow>1+2+3+…+n=\\frac{n(n+1)}{2})<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n

                \u5272\u308a\u5207\u308c\u308b\u304b\u3069\u3046\u304b\uff1a\u5f0f\u304c\u7279\u5b9a\u306e\u6570\u3067\u5272\u308a\u5207\u308c\u308b\u304b\u3069\u3046\u304b\u306e\u8a3c\u660e\uff08\u4f8b\uff1a2<\/mn>2<\/mn>n<\/mi><\/mrow><\/msup>\u2212<\/mo>1<\/mn><\/mrow>2^{2n}-1<\/annotation><\/semantics><\/math>, 2 \u306e 2 n \u4e57\u304b\u30891\u3092\u5f15\u3044\u305f\u5024\u306f3\u3067\u5272\u308a\u5207\u308c\u308b\uff09\u3002<\/p>\n\n\n\n

                \u90e8\u5206\u548c\uff1a\u548c\u306e\u516c\u5f0f\u306e\u8a3c\u660e\u3002<\/p>\n\n\n\n

                  \n
                1. \u6a19\u672c\u5272\u5408\u3068\u306e\u95a2\u4fc2\uff08\u6587\u8108\u306b\u3088\u308b\u8aa4\u89e3\uff09<\/li>\n<\/ol>\n\n\n\n

                  \u691c\u7d22\u7d50\u679c\u306b\u57fa\u3065\u304f\u3068\u3001\u300c\u6a19\u672c\u5272\u5408\u300d\uff08p\u0302\uff09\u306f\u30e6\u30cb\u30c3\u30c83\u306e\u5e30\u7d0d\u6cd5\u306e\u4e3b\u8981\u306a\u7126\u70b9\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u6a19\u672c\u5272\u5408\u306f\u3001\u65b9\u6cd5\u8ad6\uff0f\u5c02\u9580\u5206\u91ce\u306e\u7d71\u8a08\u7684\u63a8\u8ad6\u306b\u304a\u3044\u3066\u3001\u4fe1\u983c\u533a\u9593\uff08<\/p>\n\n\n\n

                  p<\/mi>\u0302<\/mo><\/mover>\u00b1<\/mo>z<\/mi>p<\/mi>\u0302<\/mo><\/mover>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>\u0302<\/mo><\/mover>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>p hat plus or minus z the square root of the fraction with numerator p hat open paren 1 minus p hat close paren and denominator n end-fraction end-root<\/annotation><\/semantics><\/math> \uff09\u3092\u7528\u3044\u3066\u6bcd\u6570\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u63a8\u5b9a\u3059\u308b\u305f\u3081\u306b\u7814\u7a76\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

                  \u300c\u5e30\u7d0d\u6cd5\u300d\u3068\u300c\u5272\u5408\u300d\u304c\u4e00\u7dd2\u306b\u51fa\u3066\u304f\u308b\u5834\u5408\u3001\u4ee5\u4e0b\u306e\u3053\u3068\u3092\u6307\u3057\u3066\u3044\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

                  nn\ud835\udc5b\u3092\u542b\u3080\u5f0f\u306e\u8a3c\u660e\uff1a\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u3001nn\uff08\u6a19\u672c\u30b5\u30a4\u30ba\uff09\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u542b\u3080\u5206\u6563\u307e\u305f\u306f\u671f\u5f85\u5024\u306e\u5f0f\u3092\u8a3c\u660e\u3059\u308b\u3002<\/p>\n\n\n\n

                    \n
                  1. \u8a66\u9a13\u306e\u4e3b\u306a\u69cb\u6210\u8981\u7d20<\/li>\n<\/ol>\n\n\n\n

                    \u30c6\u30af\u30ce\u30ed\u30b8\u30fc\uff1a\u30e6\u30cb\u30c3\u30c83\u306e\u8a66\u9a13\u3067\u306f\u3001CAS\u4ee5\u5916\u306e\u96fb\u5353\u306e\u4f7f\u7528\u304c\u901a\u5e38\u8a31\u53ef\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

                    \u30c6\u30af\u30cb\u30c3\u30af\uff1a\u8a3c\u660e\u3067\u306f\u3001k+1 \u306e\u3068\u304d\u3001\u5de6\u8fba\uff08LHS\uff09\u3068\u53f3\u8fba\uff08RHS\uff09\u304c\u7b49\u3057\u3044\u3053\u3068\u3092\u53b3\u5bc6\u306b\u793a\u3055\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n

                    \u6ce8\uff1a\u5c02\u9580\u6570\u5b66\uff08QLD\uff09\u306e\u30e6\u30cb\u30c3\u30c83\u3067\u306f\u30013\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u3001\u8907\u7d20\u6570\uff08\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\uff09\u3001\u95a2\u6570\u306b\u3064\u3044\u3066\u3082\u6271\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

                    <\/p>\n","protected":false},"excerpt":{"rendered":"

                    In QLD Year 12 Specialist Mathematics (Unit 3), Mathematical Induction is a foundational proof technique used to establish the truth of mathematical statements (propositions) for all natural numbers (n\u2208Nn is an element of the natural numbers).  While “sample proportions” are a statistical concept often found in Mathematical Methods (Topic 4, Unit 4), in the context […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1388","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1388","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1388"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1388\/revisions"}],"predecessor-version":[{"id":1455,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1388\/revisions\/1455"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1388"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1388"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1388"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}