If α , β are natural numbers such that 100 α - 199 β = ( 100 ) ( 100 ) + ( 99 ) ( 101 ) + ( 98 ) ( 102 ) + … . . + ( 1 ) ( 199 ) , then the slope of the line passing through ( α , β ) and origin is:

Step-by-step solution

Let S = ( 100 ) ( 100 ) + ( 99 ) ( 101 ) + ( 98 ) ( 102 ) + . . . + ( 2 ) ( 198 ) + ( 1 ) ( 199 ) S = ∑ x = 0 99 100 - x 100 + x = ∑ x = 0 99 100 2 - ∑ x = 0 99 x 2 S = 100 3 - 99 × 100 × 199 6 = 100 3 - 199 × 1650 Given that, S = 100 α - 199 β ∴   α = 3 , β = 1650 The slope of line passing through α , β   &   0 , 0 = 1650 - 0 3 - 0 = 550