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Algebra Difficulty 7.0 National olympiad Find the answer

8.34 The real number sequence a0,a1,a2,,an,a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots satisfies the following equation: a0=aa_{0}=a, where aa is a real number,
an=an13+13an1,nN.a_{n}=\frac{a_{n-1} \sqrt{3}+1}{\sqrt{3}-a_{n-1}}, n \in N .

Find a1994a_{1994}.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

[Solution] After calculation, we can obtain
a1=a3+13aa2=a13+13a1=(a3+1)3+(3a)3(3a)(a3+1)=a+31a3a3=a23+13a2=(a+3)3+(1a3)3(1a3)(a+3)=1a\begin{aligned} a_{1} & =\frac{a \sqrt{3}+1}{\sqrt{3}-a} \\ a_{2} & =\frac{a_{1} \sqrt{3}+1}{\sqrt{3}-a_{1}}=\frac{(a \sqrt{3}+1) \sqrt{3}+(\sqrt{3}-a)}{\sqrt{3}(\sqrt{3}-a)-(a \sqrt{3}+1)} \\ & =\frac{a+\sqrt{3}}{1-a \sqrt{3}} \\ a_{3} & =\frac{a_{2} \sqrt{3}+1}{\sqrt{3}-a_{2}}=\frac{(a+\sqrt{3}) \sqrt{3}+(1-a \sqrt{3})}{\sqrt{3}(1-a \sqrt{3})-(a+\sqrt{3})} \\ & =-\frac{1}{a} \end{aligned}

From this, we get \square
a6=11a=aa_{6}=-\frac{1}{-\frac{1}{a}}=a

It is evident that the sequence a0,a1,a2,a_{0}, a_{1}, a_{2}, \cdots is a periodic sequence with a period of 6.
Since 1994=6×332+21994=6 \times 332+2, we have
a1994=a2=a+31a3.a_{1994}=a_{2}=\frac{a+\sqrt{3}}{1-a \sqrt{3}} .

Where k,uk, u are both positive integers.
Substituting k=uu5k=u u_{5}, we get
u2u5=2000u2u5=2453\begin{array}{l} u^{2} \cdot u_{5}=2000 \\ u^{2} \cdot u_{5}=2^{4} \cdot 5^{3} \end{array}
All possible values of u2u^{2} are 1,22,24,52,2252,24521,2^{2}, 2^{4}, 5^{2}, 2^{2} \cdot 5^{2}, 2^{4} 5^{2}, thus all possible values of uu are 1, 2,22,5,25,2252,2^{2}, 5,2 \cdot 5,2^{2} \cdot 5; all possible values of kk are 2000,1000,500,400,200,1002000,1000,500,400,200,100.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.