If `x = (1 ^(2))/(1) =+ (2 ^(2))/( 3) + (3 ^(2))/( 5) +.....+ (1001 ^(2))/( 2001) , y = (1^(2))/(3) + (2 ^(2))/( 5) + (3 ^(2))/(7) + .....+ (1001 ^(2))/(2003), ` then ` ([x -y])/(10)` is equal to
where [.] denotes greatest integer function)

Options

A. 500
B. 450
C. 510
D. 555

Answer

A, B

Explanation

`x-y=1^(2)((1)/(1)-(1)/(3))+2^(2)((1)/(3)-(1)/(5))….1001^(2)((1)/(2001)-(1)/(2003))`
`=(1^(2).2)/(1.3)+(2^(2)2)/(3.5)…+(2.1001^(2))/(2001.2003)`
`(x-y)/(2)=((1^(2))/(1.3)+(2^(2))/(3.5)…(1001^(2))/(2001.2003))`
`T_(r)=(r^(2))/((2r-1)(2r+1))`
`=(1)/(4)((4r^(2)+1)/((2r-1))(-1)/((2r+1)))`
`=(1)/(4)(1+(1)/((2r-1)(2r+1)))`
`=(1)/(4)(1)+((1)/((2r-1)-(1)/((2r+1)))`
`sum_(r=1)^(1001)T_(r)=(1)/(4)(1001)+(1)/(8)(1-(1)/(2003))`
`(x-y)/(2)=(1001)/(2)+(2002)/(2003.8)`
`x-y=(1001)/(2)+(1001)/(2.2003)`
`[x-y]=500`