# Solution: 2012-19 A limit of a sequence involving a square root

Let $$a_0=3$$ and $$a_{n}=a_{n-1}+\sqrt{a_{n-1}^2+3}$$ for all $$n\ge 1$$. Determine $\lim_{n\to\infty}\frac{a_n}{2^n}.$

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-19.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3). Two incorrect solutions were submitted (YSC, KJW).

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# Good luck for your midterm exam

Good luck next week for your midterm exam! We take break and return on November 2.

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# 2012-19 A limit of a sequence involving a square root

Let $$a_0=3$$ and $$a_{n}=a_{n-1}+\sqrt{a_{n-1}^2+3}$$ for all $$n\ge 1$$. Determine $\lim_{n\to\infty}\frac{a_n}{2^n}.$

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# Solution: 2012-18 Diagonal

Let $$r_1,r_2,r_3,\ldots$$ be a sequence of all rational numbers in $$(0,1)$$ except finitely many numbers. Let $$r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots$$ be a decimal representation of $$r_j$$. (For instance, if $$r_1=\frac{1}{3}=0.333333\cdots$$, then $$a_{1,k}=3$$ for any $$k$$.)

Prove that the number $$0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots$$ given by the main diagonal cannot be a rational number.

The best solution was submitted by Kim, Joo Wan (김주완, 수리과학과 2010학번). Congratulations!

Here is his Solution of Problem 2012-18.

Alternative solutions were submitted by 이명재 (2012학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +2), 이신영 (2012학번, +2),  윤영수 (2011학번, +2), 박훈민 (대전과학고 2학년, +3), 어수강 (서울대학교 수리과학부 석사과정, +2), 윤성철 (홍익대학교 수학교육학과 2009학번, +2). There were 3 incorrect solutions submitted (JWS, KDR, JSH).

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# 2012-18 Diagonal

Let $$r_1,r_2,r_3,\ldots$$ be a sequence of all rational numbers in $$(0,1)$$ except finitely many numbers. Let $$r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots$$ be a decimal representation of $$r_j$$. (For instance, if $$r_1=\frac{1}{3}=0.333333\cdots$$, then $$a_{1,k}=3$$ for any $$k$$.)

Prove that the number $$0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots$$ given by the main diagonal cannot be a rational number.

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Let $$m$$ and $$n$$ be odd integers. Determine $\sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.$