MathDB

Special sequence

Source: St Peterburg Olympiad 2009, Grade 11, P7

August 30, 2017

algebra

Problem Statement

f(x)=x^2+x

b_1,...,b_{10000}>0

and

|b_{n+1}-f(b_n)|\leq \frac{1}{1000}

for

n=1,...,9999

Prove, that there is such

a_1>0

that

a_{n+1}=f(a_n);n=1,...,9999

and

|a_n-b_n|<\frac{1}{100}