Question

Ajutorrrr!!!Dau 99 de puncte pt. 2 exerciti!!!
1.Determinati a 2012-a zecimala a numarului 1,(234).
si
1 1 1
2.Aratati ca -------+----------+.........+--------------------<1
1 ori 2 2 ori 3 2011 ori 2012

Answer 1

$1,(234)=1,234234234234... \\ \\ Obsevam~ca~a~n-a~zecimala~este~egala~cu: \\ \\ -2,~daca~n=3k+1 \\ \\ -3,~daca~n=3k+2 \\ \\ -4,~daca~n=3k. \\ \\ 2012=3 \cdot 670+2 \Rightarrow a ~2012-a~zecimala~este~egala~cu~3. \\ \\ ------------------------------$

$Avem:~ \frac{1}{k(k+1)} = \frac{(k+1)-k}{k(k+1)}= \frac{k+1}{k(k+1)}- \frac{k}{k(k+1)}= \frac{1}{k}- \frac{1}{k+1}. \\ \\ Deci:~ \frac{1}{1 \cdot 2} =1- \frac{1}{2} \\ \\ ~~~~~~~~~~~ \frac{1}{2 \cdot 3}= \frac{1}{2}- \frac{1}{3} \\ \\ ~~~~~~~~~~~\frac{1}{3 \cdot 4} = \frac{1}{3}- \frac{1}{4} \\ \\ ~~~~~~~~~~~.................. \\ \\ ~~~~~~~~ \frac{1}{2011 \cdot 2012} = \frac{1}{2011}- \frac{1}{2012}. \\ \\ Insumand~toate~aceste~relatii,~obtinem~S=1- \frac{1}{2012},~deci~ \boxed{S\ \textless \ 1}~.$

$Cu~"S"~am~notat~suma~din~enunt.$