Răspuns:
Explicație pas cu pas:
3 + 5 + 7 + 9 + ..... + 2005 - 2 - 4 - 6 - 8 - ......- 2004 = 1002
= (3+5+7+.....+2005) - 2(1+2+3+....+1002)
(2005 - 3) : 2 + 1 = 2002:2+1 = 1002 termeni are prima suma
=> aplic formula sumei lui Gauss:
= 1002 × ( 3 + 2005) : 2 - 2 × 1002 × ( 1+ 1002) : 2 =
= 1002 × 2008/2 - 1002 × 1003 =
= 1002 × ( 1004 - 1003) =
= 1002 × 1 =
= 1002
3 + 5 + 7 + 9 + ... + 2005 - 2 - 4 - 6 - 8 - ... - 2004 = ?
(3 + 5 + 7 + 9 + ... + 2005) - (2 + 4 + 6 + 8 + ... + 2004) = 1002
[(3 + 4 + 5 + ... + 2005) - (4 + 6 + 8 + ... + 2004)] - 2 × (1 + 2 + 3 + 4 + ... + 1002) = 1002
{[2005 × (2005 + 1) ÷ 2] - [2 × (2 + 1) ÷ 2]} - 2 × (2 + 3 + 4 + ... + 1002) - 2 × [1002 × (1002 + 1) ÷ 2] = 1002
[(2005 × 2006 ÷ 2) - (2 × 3 ÷ 2)] - 2 × {[1002 × (1002 + 1) ÷ 2] - [1 × (1 + 1) ÷ 2] - 2 × (1002 × 1003 ÷ 2) = 1002
[(4022030 ÷ 2) - (6 ÷ 2)] - 2 × [(1002 × 1003 ÷ 2) - (1 × 2 ÷ 2)] - 2 × (1005006 ÷ 2) = 1002
(2011015 - 3) - 2 × [(1005006 ÷ 2) - (2 ÷ 2)] - 2 × 502503 = 1002
2011012 - 2 × (502503 - 1) - 1005006 = 1002
2011012 - 2 × 502502 - 1005006 = 1002
2011012 - 1005004 - 1005006 = 1002
1006008 - 1005006 = 1002
