AlgebraDifficulty 7.0National olympiadFind the answer
8.34 The real number sequence a0,a1,a2,⋯,an,⋯ satisfies the following equation: a0=a, where a is a real number, an=3−an−1an−13+1,n∈N.
Find a1994.
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 a1a2a3=3−aa3+1=3−a1a13+1=3(3−a)−(a3+1)(a3+1)3+(3−a)=1−a3a+3=3−a2a23+1=3(1−a3)−(a+3)(a+3)3+(1−a3)=−a1
From this, we get □ a6=−−a11=a
It is evident that the sequence a0,a1,a2,⋯ is a periodic sequence with a period of 6. Since 1994=6×332+2, we have a1994=a2=1−a3a+3.
Where k,u are both positive integers. Substituting k=uu5, we get u2⋅u5=2000u2⋅u5=24⋅53 All possible values of u2 are 1,22,24,52,22⋅52,2452, thus all possible values of u are 1, 2,22,5,2⋅5,22⋅5; all possible values of k are 2000,1000,500,400,200,100.
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