{"id":1284,"date":"2026-01-26T18:39:02","date_gmt":"2026-01-26T08:39:02","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1284"},"modified":"2026-01-27T12:20:34","modified_gmt":"2026-01-27T02:20:34","slug":"year12-math-1-2-23-growth-and-decay-in-sequences","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1284","title":{"rendered":"Year12- MATH-2-4-4 Growth and Decay in Sequences"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In QLD Year 12 Mathematics (both General Mathematics and Mathematical Methods), growth and decay are modelled using sequences to describe how a quantity changes over discrete time intervals.&nbsp;<mark><strong>Arithmetic sequences<\/strong>&nbsp;model linear changes (adding\/subtracting a constant amount), while&nbsp;<strong>geometric sequences<\/strong>&nbsp;model exponential changes (multiplying by a constant factor<\/mark>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>1. Arithmetic Sequences: Linear Growth and Decay&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Arithmetic sequences are used when a quantity increases or decreases by the <strong>same amount<\/strong> at regular time intervals.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Recursive Form:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">cap V sub n plus 1 end-sub equals cap R cap V sub n<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><strong>Explicit Rule:<\/strong> <math data-latex=\"V_{n}=V_{0}+nd\"><semantics><mrow><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><mi>n<\/mi><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{n}=V_{0}+nd<\/annotation><\/semantics><\/math> (if starting at <math data-latex=\"V_{0}\"><semantics><msub><mi>V<\/mi><mn>0<\/mn><\/msub><annotation encoding=\"application\/x-tex\">V_{0}<\/annotation><\/semantics><\/math>) or <math data-latex=\"t_{n}=t_{1}+(n-1)d\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>t<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t_{n}=t_{1}+(n-1)d<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><strong>Growth (d>0):<\/strong> The quantity increases by a constant amount (\ud835\udc51). Examples include simple interest and arithmetic progression.<\/li>\n\n\n\n<li><strong>Decay (d&lt;0):<\/strong> The quantity decreases by a constant amount (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>d<\/mi><annotation encoding=\"text\/plain\">d<\/annotation><\/semantics><\/math>). Example: Straight-line depreciation.<\/li>\n\n\n\n<li><strong>Graph:<\/strong> Linear\/straight line.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2. Geometric Sequences: Geometric Growth and Decay&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Geometric sequences model situations where a quantity increases or decreases by a <strong>constant percentage rate<\/strong> (a common ratio \ud835\udc45).&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Recursive Form:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">cap V sub n plus 1 end-sub equals cap R cap V sub n<\/annotation><\/semantics><\/math>\ud835\udc49\ud835\udc5b+1=\ud835\udc45\ud835\udc49\ud835\udc5b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Explicit Rule: <\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><msup><mi>R<\/mi><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"text\/plain\">cap V sub n equals cap V sub 0 cap R to the n-th power<\/annotation><\/semantics><\/math> (if starting at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msub><mi>V<\/mi><mn>0<\/mn><\/msub><annotation encoding=\"text\/plain\">cap V sub 0<\/annotation><\/semantics><\/math>) or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>t<\/mi><mn>1<\/mn><\/msub><msup><mi>R<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"text\/plain\">t sub n equals t sub 1 cap R raised to the n minus 1 power<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Growth (R&gt;1):<\/strong> The quantity increases exponentially. Examples: Compound interest, population growth.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Decay (0&lt;R&lt;1):<\/strong> The quantity decreases exponentially. Examples: Reducing balance depreciation, half-life.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Common Ratio (\ud835\udc45):<\/strong> Calculated as <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"text\/plain\">1 plus r<\/annotation><\/semantics><\/math> (for growth) or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"text\/plain\">1 minus r<\/annotation><\/semantics><\/math> (for decay), where<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>r<\/mi><annotation encoding=\"text\/plain\">r<\/annotation><\/semantics><\/math> is the percentage rate.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Graph:<\/strong> Exponential curve.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>3. Combined Growth and Decay (First-Order Recurrence Relations)&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In many real-world scenarios, particularly finance, both a percentage change and a flat amount change occur simultaneously.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Formula:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"text\/plain\">cap V sub n plus 1 end-sub equals cap R cap V sub n plus d<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><strong>Applications:<\/strong> This models situations like reducing balance loans where interest is added (\ud835\udc45) and repayments are made (\ud835\udc51).\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Summary Table&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><th>Feature&nbsp;<\/th><th>Arithmetic (Linear)<\/th><th>Geometric (Exponential)<\/th><\/tr><tr><td><strong>Change Type<\/strong><\/td><td>Constant amount (\ud835\udc51)<\/td><td>Constant percentage ratio (\ud835\udc45)<\/td><\/tr><tr><td><strong>Recursive Rule<\/strong><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"text\/plain\">cap V sub n plus 1 end-sub equals cap V sub n plus d<\/annotation><\/semantics><\/math><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">cap V sub n plus 1 end-sub equals cap R cap V sub n<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td><strong>Growth Condition<\/strong><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">d is greater than 0<\/annotation><\/semantics><\/math><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><mo>&gt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">cap R is greater than 1<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td><strong>Decay Condition<\/strong><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">d is less than 0<\/annotation><\/semantics><\/math><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>0<\/mn><mo>&lt;<\/mo><mi>R<\/mi><mo>&lt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">0 is less than cap R is less than 1<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td><strong>Graph Shape<\/strong><\/td><td>Straight line<\/td><td>Exponential curve<\/td><\/tr><tr><td><strong>Examples<\/strong><\/td><td>Simple interest, straight-line depreciation<\/td><td>Compound interest, reducing balance depreciation<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Applications in QLD Year 12&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>General Math:<\/strong> Focuses heavily on using these for financial modelling (simple\/compound interest, depreciation, annuities).<\/li>\n\n\n\n<li><strong>Math Methods:<\/strong> Focuses on the formulas, graphing, and understanding the limiting behaviour (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mo>\u2192<\/mo><mo>\u221e<\/mo><\/mrow><annotation encoding=\"text\/plain\">n right arrow infinity<\/annotation><\/semantics><\/math>) of sequences.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">****************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u306e\u6570\u5b66\uff08\u4e00\u822c\u6570\u5b66\u3068\u6570\u5b66\u7684\u624b\u6cd5\u306e\u4e21\u65b9\uff09\u3067\u306f\u3001\u6570\u91cf\u304c\u96e2\u6563\u7684\u306a\u6642\u9593\u9593\u9694\u3067\u3069\u306e\u3088\u3046\u306b\u5909\u5316\u3059\u308b\u304b\u3092\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306b\u3001\u6570\u5217\u3092\u7528\u3044\u3066\u5897\u52a0\u3068\u6e1b\u5c11\u3092\u30e2\u30c7\u30eb\u5316\u3057\u307e\u3059\u3002\u7b49\u5dee\u6570\u5217\u306f\u7dda\u5f62\u5909\u5316\uff08\u4e00\u5b9a\u91cf\u306e\u52a0\u7b97\u307e\u305f\u306f\u6e1b\u7b97\uff09\u3092\u30e2\u30c7\u30eb\u5316\u3057\u3001\u7b49\u5dee\u6570\u5217\u306f\u6307\u6570\u5909\u5316\uff08\u4e00\u5b9a\u4fc2\u6570\u306e\u4e57\u7b97\uff09\u3092\u30e2\u30c7\u30eb\u5316\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u7b49\u5dee\u6570\u5217\uff1a\u7dda\u5f62\u5897\u52a0\u3068\u7dda\u5f62\u6e1b\u5c11<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u7b49\u5dee\u6570\u5217\u306f\u3001\u6570\u91cf\u304c\u4e00\u5b9a\u306e\u6642\u9593\u9593\u9694\u3067\u540c\u3058\u91cf\u3060\u3051\u5897\u52a0\u307e\u305f\u306f\u6e1b\u5c11\u3059\u308b\u5834\u5408\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<br>\u518d\u5e30\u5f62\u5f0f: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">cap V sub n plus 1 end-sub equals cap R cap V sub n<\/annotation><\/semantics><\/math><br>\u660e\u793a\u7684\u898f\u5247: <math data-latex=\"V_{n}=V_{0}+nd\"><semantics><mrow><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><mi>n<\/mi><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{n}=V_{0}+nd<\/annotation><\/semantics><\/math> (<math data-latex=\"V_{0}\"><semantics><msub><mi>V<\/mi><mn>0<\/mn><\/msub><annotation encoding=\"application\/x-tex\">V_{0}<\/annotation><\/semantics><\/math> \u304b\u3089\u958b\u59cb\u3059\u308b\u5834\u5408) \u307e\u305f\u306f <math data-latex=\" t_{n}=t_{1}+(n-1)d\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>t<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t_{n}=t_{1}+(n-1)d<\/annotation><\/semantics><\/math><br>\u5897\u52a0 <math data-latex=\"(d&gt;0)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>d<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(d&gt;0)<\/annotation><\/semantics><\/math>: \u6570\u91cf\u306f\u4e00\u5b9a\u91cf <math data-latex=\"(d)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>d<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(d)<\/annotation><\/semantics><\/math> \u3060\u3051\u5897\u52a0\u3057\u307e\u3059\u3002\u4f8b\u3068\u3057\u3066\u306f\u3001\u5358\u5229\u3084\u7b49\u5dee\u6570\u5217\u306a\u3069\u304c\u3042\u308a\u307e\u3059\u3002<br>\u6e1b\u5c11 <math data-latex=\"(d<0)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>d<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(d&lt;0)<\/annotation><\/semantics><\/math>: \u6570\u91cf\u306f\u4e00\u5b9a\u91cf (<math data-latex=\"d\"><semantics><mi>d<\/mi><annotation encoding=\"application\/x-tex\">d<\/annotation><\/semantics><\/math>) \u3060\u3051\u6e1b\u5c11\u3057\u307e\u3059\u3002\u4f8b: \u5b9a\u984d\u6e1b\u4fa1\u511f\u5374\u3002<br>\u30b0\u30e9\u30d5: \u76f4\u7dda\u3002<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>\u5e7e\u4f55\u7d1a\u6570: \u5e7e\u4f55\u7d1a\u6570\u7684\u5897\u52a0\u3068\u5e7e\u4f55\u7d1a\u6570\u7684\u6e1b\u5c11<br>\u5e7e\u4f55\u7d1a\u6570\u306f\u3001\u6570\u91cf\u304c\u4e00\u5b9a\u306e\u30d1\u30fc\u30bb\u30f3\u30c6\u30fc\u30b8 (\u516c\u6bd4 (<math data-latex=\"R\"><semantics><mi>R<\/mi><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math>)) \u3060\u3051\u5897\u52a0\u307e\u305f\u306f\u6e1b\u5c11\u3059\u308b\u72b6\u6cc1\u3092\u30e2\u30c7\u30eb\u5316\u3057\u307e\u3059\u3002<br>\u518d\u5e30\u5f62\u5f0f: (<math data-latex=\"V_{n+1}=RV_{n}\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{n+1}=RV_{n}<\/annotation><\/semantics><\/math>)<br>\u660e\u793a\u7684\u306a\u898f\u5247: (<math data-latex=\"V_{n}=V_{0}R^{n}\"><semantics><mrow><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><msup><mi>R<\/mi><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">V_{n}=V_{0}R^{n}<\/annotation><\/semantics><\/math>) (<math data-latex=\"V_{0}\"><semantics><msub><mi>V<\/mi><mn>0<\/mn><\/msub><annotation encoding=\"application\/x-tex\">V_{0}<\/annotation><\/semantics><\/math> \u304b\u3089\u958b\u59cb\u3059\u308b\u5834\u5408) \u307e\u305f\u306f                             (<math data-latex=\"t_{n}=t_{1}R^{n-1}\"><semantics><mrow><msub><mi>t<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>t<\/mi><mn>1<\/mn><\/msub><msup><mi>R<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">t_{n}=t_{1}R^{n-1}<\/annotation><\/semantics><\/math>)<br>\u5897\u52a0 <math data-latex=\"(R&gt;1)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo>&gt;<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(R&gt;1)<\/annotation><\/semantics><\/math>: \u6570\u91cf\u306f\u6307\u6570\u7684\u306b\u5897\u52a0\u3057\u307e\u3059\u3002\u4f8b: \u8907\u5229\u3001\u4eba\u53e3\u5897\u52a0\u3002<br>\u6e1b\u8870 (<math data-latex=\"0&lt;R&lt;1\"><semantics><mrow><mn>0<\/mn><mo>&lt;<\/mo><mi>R<\/mi><mo>&lt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">0&lt;R&lt;1<\/annotation><\/semantics><\/math>): \u6570\u91cf\u306f\u6307\u6570\u7684\u306b\u6e1b\u5c11\u3057\u307e\u3059\u3002\u4f8b: \u9013\u6e1b\u5747\u8861\u3001\u534a\u6e1b\u671f\u3002<br>\u516c\u6bd4 <math data-latex=\"(R)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(R)<\/annotation><\/semantics><\/math>: \u5897\u52a0\u306e\u5834\u5408\u306f <math data-latex=\"(1+r)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>r<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(1+r)<\/annotation><\/semantics><\/math>\u3001\u6e1b\u8870\u306e\u5834\u5408\u306f<math data-latex=\" (1-r) \"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>r<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\"> (1-r) <\/annotation><\/semantics><\/math>\u3068\u3057\u3066\u8a08\u7b97\u3055\u308c\u307e\u3059\u3002(<math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>) \u306f\u30d1\u30fc\u30bb\u30f3\u30c8\u3067\u3059\u3002<br>\u30b0\u30e9\u30d5: \u6307\u6570\u66f2\u7dda\u3002<\/li>\n\n\n\n<li>\u5897\u52a0\u3068\u6e1b\u5c11\u306e\u7d44\u307f\u5408\u308f\u305b\uff08\u4e00\u6b21\u56de\u5e30\u95a2\u4fc2\uff09<br>\u591a\u304f\u306e\u73fe\u5b9f\u4e16\u754c\u306e\u30b7\u30ca\u30ea\u30aa\u3001\u7279\u306b\u91d1\u878d\u306e\u4e16\u754c\u3067\u306f\u3001\u30d1\u30fc\u30bb\u30f3\u30c6\u30fc\u30b8\u306e\u5909\u5316\u3068\u4e00\u5b9a\u984d\u306e\u5909\u5316\u304c\u540c\u6642\u306b\u767a\u751f\u3057\u307e\u3059\u3002<br>\u5f0f\uff1a(<math data-latex=\"V_{n+1}=RV_{n}+d\"><semantics><mrow><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{n+1}=RV_{n}+d<\/annotation><\/semantics><\/math>)<br>\u5fdc\u7528\uff1a\u3053\u308c\u306f\u3001\u5229\u5b50 <math data-latex=\"(R)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(R)<\/annotation><\/semantics><\/math>\u304c\u52a0\u7b97\u3055\u308c\u3001\u8fd4\u6e08 (<math data-latex=\"d)\"><semantics><mrow><mi>d<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">d)<\/annotation><\/semantics><\/math> \u304c\u884c\u308f\u308c\u308b\u3001\u6b8b\u9ad8\u6e1b\u5c11\u578b\u30ed\u30fc\u30f3\u306a\u3069\u306e\u72b6\u6cc1\u3092\u30e2\u30c7\u30eb\u5316\u3057\u307e\u3059\u3002<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u8981\u7d04\u8868<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>\u7279\u5fb4<\/td><td>\u7b97\u8853\uff08\u7dda\u5f62\uff09<\/td><td>\u5e7e\u4f55\uff08\u6307\u6570\uff09<\/td><\/tr><tr><td>\u5909\u5316\u306e\u7a2e\u985e<\/td><td>\u5b9a\u984d\uff08(d)\uff09<\/td><td>\u5b9a\u7387\uff08(R)\uff09<\/td><\/tr><tr><td>\u518d\u5e30\u898f\u5247<\/td><td><math data-latex=\"(V_{n+1}=V_{n}+d)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><mi>d<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(V_{n+1}=V_{n}+d)<\/annotation><\/semantics><\/math><\/td><td><math data-latex=\"(V_{n+1}=RV_{n})\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mi>R<\/mi><msub><mi>V<\/mi><mi>n<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(V_{n+1}=RV_{n})<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td>\u5897\u52a0\u6761\u4ef6<\/td><td><math data-latex=\"(d&gt;0)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>d<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(d&gt;0)<\/annotation><\/semantics><\/math><\/td><td><math data-latex=\"(R&gt;1)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo>&gt;<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(R&gt;1)<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td>\u6e1b\u5c11\u6761\u4ef6<\/td><td><math data-latex=\"(d&lt;0)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>d<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(d&lt;0)<\/annotation><\/semantics><\/math><\/td><td><math data-latex=\"(0&lt;R&lt;1)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>&lt;<\/mo><mi>R<\/mi><mo>&lt;<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(0&lt;R&lt;1)<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td>\u30b0\u30e9\u30d5\u306e\u5f62\u72b6<\/td><td>\u76f4\u7dda<\/td><td>\u6307\u6570\u66f2\u7dda<\/td><\/tr><tr><td>\u4f8b\uff1a<\/td><td>\u5358\u5229\u3001\u5b9a\u984d\u6e1b\u4fa1\u511f\u5374\u3001<\/td><td>\u8907\u5229\u3001\u9013\u6e1b\u6cd5\u5b9a\u984d\u6e1b\u4fa1\u511f\u5374<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u306b\u304a\u3051\u308b\u5fdc\u7528<br>\u4e00\u822c\u6570\u5b66\uff1a\u3053\u308c\u3089\u3092\u8ca1\u52d9\u30e2\u30c7\u30ea\u30f3\u30b0\uff08\u5358\u5229\uff0f\u8907\u5229\u3001\u6e1b\u4fa1\u511f\u5374\u3001\u5e74\u91d1\uff09\u306b\u6d3b\u7528\u3059\u308b\u3053\u3068\u306b\u91cd\u70b9\u3092\u7f6e\u304d\u307e\u3059\u3002<br>\u6570\u5b66\u7684\u624b\u6cd5\uff1a\u6570\u5217\u306e\u516c\u5f0f\u3001\u30b0\u30e9\u30d5\u4f5c\u6210\u3001\u304a\u3088\u3073\u6975\u9650\u6319\u52d5 (<math data-latex=\"n\\rightarrow \\infty \"><semantics><mrow><mi>n<\/mi><mo stretchy=\"false\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n\\rightarrow \\infty <\/annotation><\/semantics><\/math>) \u306e\u7406\u89e3\u306b\u91cd\u70b9\u3092\u7f6e\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In QLD Year 12 Mathematics (both General Mathematics and Mathematical Methods), growth and decay are modelled using sequences to describe how a quantity changes over discrete time intervals.&nbsp;Arithmetic sequences&nbsp;model linear changes (adding\/subtracting a constant amount), while&nbsp;geometric sequences&nbsp;model exponential changes (multiplying by a constant factor). 1. Arithmetic Sequences: Linear Growth and Decay&nbsp; Arithmetic sequences are used [&hellip;]<\/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-1284","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1284","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=1284"}],"version-history":[{"count":4,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1284\/revisions"}],"predecessor-version":[{"id":1368,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1284\/revisions\/1368"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1284"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1284"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1284"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}