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a_{n+1} =(a_1 + a_2 + ...+ a_n)/n+1, for every b exists a_k : a_k < bk

Source: Norwegian Mathematical Olympiad 2013 - Abel Competition p1b

September 4, 2019

Sequencealgebrainequalities

Problem Statement

The sequence

a_1, a_2, a_3,...

is defined so that

a_1 = 1

and

a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1

for

n \ge 1

. Show that for every positive real number

b

we can find

a_k

so that

a_k < bk