Question

What is the value of 

∑_{1≤i<j≤10,i+j =odd} (i - j) - ∑_{1≤i<j≤10,i+j =even}  (i - j) ?

Answer 1

i = 1 => 1 + (2 + 4 + 6 + 8 + 10 – 3 – 5 – 7 – 9)

= 1 – 4 + 10 = 7

i = 2 => 0 × 2 + (3 + 5 + 7 + 9 – 4 – 6 – 8 – 10)

= – 4

i = 3 => 1 × 3 + (4 + 6 + 8 + 10 – 5 – 7 – 9)

= 3 – 3 + 10 = 10

i = 4 => 0 × 4 + (5 + 7 + 9 – 6 – 8 – 10) = – 3

i = 5 => 1 × 5 + (6 + 8 + 10 – 7 – 9) = 5 – 2 + 10

= 13

i = 6 => 0 × 6 + (7 + 9 – 8 – 10) = – 2

i = 7 => 1 × 7 + (8 + 10 – 9) = 7 – 1 + 10 = 16

i = 8 => 0 × 8 + (9 – 10) = – 1

i = 9 => 1 × 9 + (10) = 19

∴ S = (7 + 10 + 13 + 16 + 19) – (4 – 3 – 2 – 1)

= 55