HMMT 二月 2012 · 代数 · 第 6 题
HMMT February 2012 — Algebra — Problem 6
题目详情
英文原题
Let a0=−2,b0= 1, and for n≥0, let
an+1=an+bn+/radicalbig
a2n+b2n,
bn+1=an+bn−/radicalbig
a2n+b2n.
Find a2012.
解析
英文解析
- Let a = − 2, b = 1, and for n ≥ 0, let
0 0
√
2 2
a = a + b + a + b ,
n +1 n nn n
√
2 2
b = a + b − a + b .
n +1 n nn n
Find a .
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√
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Answer: 2 2 + 2 − 2 We have a + b = 2( a + b )
n +1 n +1 n n
2 2 2
a b = ( a + b ) − a − b = 2 a bn +1 n +1 n n n nn n
Thus,
a + b = − 2 nn nn +1
a b = − 2
n n
Using Viete’s formula, a and b are the roots of the following quadratic, and, since the square
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root is positive, a is the bigger root:
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2 2012 2013
x + 2 x − 2
Thus,
√
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2010
a = 2 2 + 2 − 2
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