https://blog.js7777.party/categories/%EC%88%98%ED%95%99/Recent content in 수학 on 이왓슨 작업실Hugoko-KRThu, 04 Dec 2025 20:14:33 +0900https://blog.js7777.party/posts/120/Thu, 04 Dec 2025 20:14:33 +0900https://blog.js7777.party/posts/120/<p>우리가 평소에 보는 대부분의 수열은 계차수열이거나, 일차 선형 점화식으로 유도된다.</p> <p>그중 일차 선형 점화식은 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>r</mi><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mi>d</mi><mspace width="1em"/><mo stretchy="false">(</mo><mi>r</mi><mo mathvariant="normal">≠</mo><mn>1</mn><mo separator="true">,</mo><mi>d</mi><mo mathvariant="normal">≠</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_{n+1} = r a_n + d \quad (r \neq 1, d \neq 0)</annotation></semantics></math></span> 꼴을 의미한다.</p> <p>일반항을 유도하기 위한 아이디어는 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>k</mi><mo>=</mo><mi>r</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_{n+1} + k = r(a_n + k)</annotation></semantics></math></span>를 만족하는 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span>를 찾는 것이다. 즉, 수열 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span>를 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span>만큼 평행이동시켜서 새로운 등비수열 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">a_n + k</annotation></semantics></math></span>를 만드는 것이다.</p>https://blog.js7777.party/posts/9/Thu, 02 Jan 2025 23:45:46 +0900https://blog.js7777.party/posts/9/<p>2025년이 시작되었다. 아쉬웠던 2024년의 기억은 털어버리고 발전을 다짐하며, 2025에 대한 수학적 고찰들을 가져왔다.</p> <p>2025년은 아마도 우리가 맞이할 수 있는 유일한 제곱수의 해이다. 다음 제곱수의 해는 91년을 더 기다려야 하기 때문이다.</p>https://blog.js7777.party/posts/8/Thu, 21 Nov 2024 00:26:55 +0900https://blog.js7777.party/posts/8/<figure class="img"><img src="https://blog.js7777.party/posts/8/img-01.png" alt="" loading="lazy" decoding="async" width="818" height="603"></figure><p>직각삼각형 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">ABC</annotation></semantics></math></span>에서, 내접원과 빗변의 교점을 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span>라고 하자. 직각삼각형 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">ABC</annotation></semantics></math></span>의 넓이 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span>에 대한 다음 식이 항상 성립한다.</p> <div class="math-block"><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>F</mi></mrow><mo stretchy="true">‾</mo></mover><mo>×</mo><mover accent="true"><mrow><mi>F</mi><mi>C</mi></mrow><mo stretchy="true">‾</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover><mo>×</mo><mover accent="true"><mrow><mi>B</mi><mi>C</mi></mrow><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>S</mi></mrow><annotation encoding="application/x-tex"> \overline{AF} \times \overline{FC} = \frac{1}{2} \times \overline{AB} \times \overline{BC} = S </annotation></semantics></math></span></div><h3 id="증명">증명<a class="hanchor" href="#%ec%a6%9d%eb%aa%85" aria-label="이 제목 링크 복사">#</a></h3><p>TODO.</p>https://blog.js7777.party/posts/7/Thu, 21 Nov 2024 00:20:05 +0900https://blog.js7777.party/posts/7/<figure class="img"><img src="https://blog.js7777.party/posts/7/img-01.png" alt="" loading="lazy" decoding="async" width="666" height="428"></figure><p>삼각형 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">ABC</annotation></semantics></math></span>에서 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>C</mi></mrow><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><mover accent="true"><mrow><mi>B</mi><mi>C</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{AC}, \overline{BC}</annotation></semantics></math></span>를 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>:</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2:1</annotation></semantics></math></span>로 내분하는 점 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo separator="true">,</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">L, H</annotation></semantics></math></span>가 있다.</p> <p><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>H</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{AH}</annotation></semantics></math></span>와 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>B</mi><mi>L</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{BL}</annotation></semantics></math></span>의 교점을 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span>이라고 하면, <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>A</mi><mi>M</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">\Delta AMB</annotation></semantics></math></span>의 넓이는 항상 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>A</mi><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">\Delta ABC</annotation></semantics></math></span>의 넓이의 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{2}</annotation></semantics></math></span>이다.</p> <h3 id="증명">증명<a class="hanchor" href="#%ec%a6%9d%eb%aa%85" aria-label="이 제목 링크 복사">#</a></h3><p><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover><mo>∥</mo><mover accent="true"><mrow><mi>L</mi><mi>H</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{AB} \parallel \overline{LH}</annotation></semantics></math></span>이므로 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>A</mi><mi>B</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">\Delta ABM</annotation></semantics></math></span>과 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>H</mi><mi>L</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">\Delta HLM</annotation></semantics></math></span>은 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mo>:</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3:1</annotation></semantics></math></span> 닮음이다.</p>