MathDB

Math Prize 2012 Problem 19

Source:

September 24, 2012

functioncalculusderivativeinduction

Problem Statement

Define

L(x) = x - \frac{x^2}{2}

for every real number

x

. If

n

is a positive integer, define

a_n

by

a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),

where there are

n

iterations of

L

. For example,

a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).

As

n

approaches infinity, what value does

n a_n

approach?

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