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수학동아 - 폴리매스
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[잡담] 산술기하평균에 관한 고찰
kty 2022.04.01 10:11 조회 238
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

수학 문제를 풀면서 산기를 써서 최대최소 구하는 것을 연습하다 문득 생각난 내용입니다.

 

어떤 두 수, a와 x를 정한 다음 a와 x의 산술평균과 기하평균의 산술평균과 기하평균의 산술평균과 기하평균의 산술평균과 기하평균의 산술평균과 기하평균..........을 반복하면 반드시 어느 한 값에 수렴하게 될 것입니다. 예를 들어 a=1, x=3이면 그 값은 1.86............에 한없이 가까워지게 됩니다.

 

이 함수를 산술과 기하를 따서 sga(x)라 하면, sga(x)의 수식을 대수적으로 어떻게 표현할 수 있을까 몇십 분 정도를 가만히 생각해 보았으나 영 답이 안 나와서 여러분들에게 이를 물어보고자 합니다. 여기는 능력자 많으니까 혹시 아시는 분이 있을 수도 있잖아요.

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