請計算 3- 4 + 5 - 6 +L+ 20......

S1 = 3 + 5 + 7 + ......... + 2005 + 2007 是等差級數的和

T(1) = a = 3
T(n) = l = 2007
d = 2

T(n) = a + (n - 1)d
2007 = 3 + (n - 1)(2)
n = 1003

S1
= n(a + l)/2
= 1003(3 + 2007)/2
= 1008015



S2 = 4 + 6 + 8 + ......... + 2004 + 2006 是等差級數的和

T(1) = a = 4
T(n) = l = 2006
d = 2

T(n) = a + (n - 1)d
2006 = 4 + (n - 1)(2)
n = 1002

S2
= n(a + l)/2
= 1002(4 + 2006)/2
= 1007010



3 - 4 + 5 - 6 +......+ 2003 - 2004 + 2005 - 2006 + 2007
= (3 + 5 + 7 + ..... + 2007) - (4 + 6 + 8 + ...... + 2006)
= S1 - S2
= 1008015 - 1007010
= 1005
=

3 - 4 + 5 - 6 +......+ 2003- 2004 + 2005 - 2006 + 2007
2007 - 2006 + 2005 - 2004 + ....... + 7 - 6 + 5 - 4 + 3
--------------------------------------------------------------------
兩式相加
= 2010 - 2010 + 2010 - 2010 + ........... + 2010 - 2010 + 2010
= 2010

所以
3- 4 + 5 - 6 +......+ 2003- 2004 + 2005 - 2006 + 2007
= 2010 / 2
= 1005

3- 4 + 5 - 6 +......+ 2003- 2004 + 2005 - 2006 + 2007
=(3- 4) + (5 - 6) +......+ (2003- 2004) + (2005 - 2006) + 2007
=(-1) x 1/2 (2006 - 4) + 2007
=(-1) x 1001 + 2007
=-1001 + 2007
=1006

其實ja係=

(-1)+(-1)+(-1).........+(-1)

e.g

Find 1-2=3-4=5-6=7-8.....+9999-10000

= (-1)+(-1)+(-1)

= (-1)X5000 = -5000