Question

23 Efectuați: a 7+9+11+...+2011-6-8-10-...-2010; b1+2-3+4+5-6+7+8-9+...+2008 +2009 - 2010 + 2011; c1+2+3-4+5+6+7-8+...+2009 +2010 +2011 – 2012. va rog repedeeeeeeeeeeeeeeee dau coroana,5puncte si inima va roooog frumos ​

Answer 1

Explicație pas cu pas:

a)

$7+9+11+...+2011-6-8-10-...-2010 = (7 - 6) + (9 - 8) + (11 - 10) + ... + (2011 - 2010) = \underbrace{1 + 1 + 1 + ... + 1}_{1003 \ \ termeni} = \bf 1003$

b)

$1+2-3+4+5-6+7+8-9+...+2008 +2009 - 2010 + 2011 = (1 + 2 - 3) + (4 + 5 - 6) + (7 + 8 - 9) + (10 + 11 - 12) + ... + (2008 + 2009 - 2010) + 2011 = 0 + 3 + 6 + 9 + ... + 2007 + 2011 = 3 \cdot (1 + 2 + 3 + ... + 669) + 2011 = \dfrac{669 \cdot 670}{2} + 2011 = \bf 226126$

c)

$1+2+3-4+5+6+7-8+...+2009 +2010 +2011 – 2012 = (1 + 2 + 3 - 4) + (5 + 6 + 7 - 8) + (9 + 10 + 11 - 12) + ... + (2009 + 2010 + 2011 - 2012) = 2 + 10 + 18 + ... + 4018 = 2 + (2 + 8) + (2 + 16) + ... + (2 + 4016) = \underbrace{2 + 2 + 2 + ... + 2}_{503 \ \ termeni} + (8 + 16 + ... + 4016) = 2 \cdot 503 + 8 \cdot (1 + 2 + ... + 502) = 1006 + 8 \cdot \dfrac{502 \cdot 503}{2} = 1006 + 1010024 = \bf 1011030$