Question

Efectuati prin Suma Gauss ​

Answer 1

Răspuns:

b) 674356 c) 1011030

Explicație pas cu pas:

b)

Notez

S=1+2-3+4+5-6+7+8-9+....+2008+2009-2010+2011 =>

S-1=2-3+4+5-6+7+8-9+....+2008+2009-2010+2011

S-1=2011-2010+2009+2008-2007+2006+2005-2004+....+5+4-3+2

2·(S-1)=(2+2011)-(3+2010)+(4+2009)+(5+2008)-(6+2007)+(7+2006)+(8+2005)-(9+2004)+...+(2008+5)+(2009+4)-(2010+3)+(2011+2)=

=2013-2013+2013+2013-2013+2013+2013-2013+...+2013+2013-2013+2013

Intrucat aceasta suma are 2010 termeni, iar dupa efectuarea scaderilor ramane tot al treilea termen => raman 2010/3=670 termeni de 2013 =>

2·(S-1)=2013·670 => S-1=2013·335 => S=2013·335+1=674356

c)

Notez

S=1+2+3-4+5+6+7-8+....+2009+2010+2011-2012 =>

S+2012=1+2+3-4+5+6+7-8+....+2007-2008+2009+2010+2011

S+2012=2011+2010+2009-2008+2007+2006+2005-2004+...+5-4+3+2+1

2·(S+2012)=(1+2011)+(2+2010)+(3+2009)-(4+2008)+(5+2007)+(6+2006)+(7+2005)-(8+2004)+...+(2007+5)-(2008+4)+(2009+3)+(2010+2)+(2011+1)=

=2012+2012+2012-2012+2012+2012+2012-2012+... +2012+2012+2012-2012+2012+2012+2012

Intrucat aceasta suma are 2011 termeni, iar dupa efectuarea scaderilor raman jumatate din primii 2008 termeni plus ultimii 3 termeni=>

2·(S+2012)=2012·2008/2+3·2012 => 2·S=2012·2008/2+3·2012-2·2012 => 2·S=2012·1004+2012=2012·(1004+1)=2012·1005 =>

S=2012·1005/2=1006·1005=1011030