Question

1)Calculati
1+3+5+........+2006=?
AJUTATI-MA VA ROOOOOOOOG!

Answer 1

Screim suma astfel:

$S=(2*0+1)+(2*1+1)+(2*2+1)+.......+(2*1002+1)$

Dam factor comun pe 2 si observam ca avem in total 1005 termeni deci apare 1 de 1005 ori in acele paranteze:

$S= 2(0+1+2+3+4+...........+1004) +1005$

Avem formula  $1+2+3+4+.......+n= \frac{n*(n+1)}{2}$

Si o folosim pentru suma noastra:

$S= 2*\frac{1004*1005}{2} +1005$

$S=504410 + 1005=505515$

Answer 2

   1 = 2·0 + 1
   3 = 2·1 + 1
   5 = 2·2 + 1
------------------
2005 = 2·1002 + 1 ⇒
⇒ S1 = 1+3+5+.......+2005 = 2(0+1+2+......1002) + 1· 1003 =
=  2·1002·1003 /2 + 1003 = 1003(1002+1) = 1003²
S1 + 2006 = 1003(1003 + 2) = 1003·1005 = 1008015