Question

calculează - 1009 +(2-4+6-8+..2014-2016 +2018 - 2020)​

Answer 1

=-1009+(2+6+10+...+2014+2018-4-8-12-...-2016-2020)=

=-1009+2*(1+3+5+...+1007+1009)-4*(1+2+3+...+504+505)=

(1+3+..1009)->progr aritmetica de ratie 2

formula: ((a1+an)*n)/2 a1 este primul termen

an este ultimul

r=ratie=a2-a1=3-1=2

an=a1+(n-1)r=>

1009=1+(n-1)2

1009=1+2n-2

1009=2n-1

2n=1010

n=505

1+2+3....+505->suma lui Gauss

formula an(an+1)/2

an este ultimul termen adica 505

=-1009+2(1+1009)*505/2-4*505*506/2=

=-1009+2*1010*505/2-4*505*253=

=-1009+2*505*505-4*505*253=

=-1009+505(2*505-4*253)=

=-1009+505(1010-1012)=

=-1009+505*(-2)=

=-1009-1010=

=-2019