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Algebra Difficulty 4.6 AIME Prove it

Given the sequence {an}\{a_n\} satisfies a1=14a_1 = \frac {1}{4}, and (1an)an+1=14,(1-a_n)a_{n+1}= \frac {1}{4},
(1) Prove that the sequence {1an12}\left\{ \frac {1}{a_n - \frac{1}{2}} \right\} is an arithmetic sequence;
(2) Prove that a2a1+a3a2++an+1an<n+34.\frac {a_2}{a_1} + \frac {a_3}{a_2} + \ldots + \frac {a_{n+1}}{a_n} < n+ \frac {3}{4}.

Solution

(1) Let bn=1an12.b_n = \frac {1}{a_n - \frac{1}{2}}. Thus, we have an=1bn+12.a_n = \frac {1}{b_n} + \frac {1}{2}.
Since (1an)an+1=14,(1-a_n)a_{n+1}= \frac {1}{4}, then:
(1(1bn+12))(1bn+1+12)=14.(1- \left( \frac {1}{b_n} + \frac {1}{2} \right)) \left( \frac {1}{b_{n+1}} + \frac {1}{2} \right) = \frac {1}{4}.
This simplifies to:
bn+1bn=2.b_{n+1} - b_n = -2.
Given that b1=11412=4,b_1 = \frac {1}{ \frac{1}{4} - \frac{1}{2}} = -4,
we can deduce that {bn}\{b_n\} is an arithmetic sequence with a first term of 4-4 and a common difference of 2.-2. Hence, the sequence {1an12}\left\{ \frac {1}{a_n - \frac{1}{2}} \right\} is an arithmetic sequence.

(2) Knowing from (1) that bn=42(n1)=2n2,b_n = -4 - 2(n-1) = -2n - 2,
we get an=1bn+12=12n+2+12=n2(n+1).a_n = \frac {1}{b_n} + \frac {1}{2} = - \frac {1}{2n + 2} + \frac {1}{2} = \frac {n}{2(n+1)}. Therefore:
an+1an=n+12(n+2)2(n+1)n=(n+1)2n(n+2)=1+1n(n+2)=1+12(1n1n+2).\frac {a_{n+1}}{a_n} = \frac {n+1}{2(n+2)} \cdot \frac {2(n+1)}{n} = \frac {(n+1)^2}{n(n+2)} = 1+ \frac {1}{n(n+2)} = 1+ \frac {1}{2}\left( \frac {1}{n} - \frac {1}{n+2} \right).
Summing this from 11 to nn, we find:
a2a1+a3a2++an+1an=n+12(113+1214++1n1n+2).\frac {a_2}{a_1} + \frac {a_3}{a_2} + \ldots + \frac {a_{n+1}}{a_n} = n+ \frac {1}{2}\left(1 - \frac {1}{3} + \frac {1}{2} - \frac {1}{4} + \ldots + \frac {1}{n} - \frac {1}{n+2}\right).
The series telescopes to:
n+12(1+121n+11n+2),n+ \frac {1}{2}\left(1+ \frac {1}{2} - \frac {1}{n+1} - \frac {1}{n+2}\right),
which is less than n+34.n+ \frac {3}{4}. Hence we have \boxed{n+ \frac {1}{2}\left(1+ \frac {1}{2} - \frac {1}{n+1} - \frac {1}{n+2}\right) < n+ \frac {3}{4}}.

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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.