Épreuve de setion européenne
A simple sequene not given by a simple formula
Let
a 1,a 2,a 3,...berealnumbersgeneratedbythefollowingrule.Therstnumber,a 1,equals
1,andthereaftereahnumberequalsthepreviousnumberplusitssquareroot.Thissimplystated
a 3,...berealnumbersgeneratedbythefollowingrule.Therstnumber,a 1,equals
1,andthereaftereahnumberequalsthepreviousnumberplusitssquareroot.Thissimplystated
rule raisesan obvious question :what is thenumber
a n?
To get a feel for the question, let us work out
a n for a few small values of n
. We have
a 2 = 1 + √
1 = 2
.Thena 3 = 2 + √ 2 a 4 = 2 + √
2 + q
2 + √ 2
andsoon.Notiethattheexpressionsontheright-handsidedonotseemtosimplify,andthat
eah new one is twie as longas theprevious one. From this observation it follows quiteeasily
that theexpressionfor
a 12 wouldinvolve 1024 ourenes ofthenumber 2,most ofthem inside
ajungleofsquareroots.Suhanexpressionwouldnot giveus muhinsightintothenumbera 12.
Adaptedfrom Mathematis, AVery ShortIntrodutionbyTimothyGowers
Questions
1. Translatethe denition ofthesequene
a n asa relationbetween a n+1 and a n.
a n.
2. Givetherawformulafor
a 5.
3. Explain the sentene Notie that the expressions on the right-hand side do not seem to
simplify,and thateahnew one istwieaslongastheprevious one.
4. Prove thatthe expression of
a 12
wouldinvolve 1024 ourenesofthe number 2.
5. Theaim ofthe following questionsistoount thenumberof squareroot signsin
a n.Let's
denote thisnumber byu n,for any naturalnumber n
.
n
.a. Find out the valuesof
u n for n
from 1to 5.
b. Prove byindutionthat
u n = 2 n − 1 − 1
.. Use theformulato ompute thenumberof squareroot signsin
