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Statement -1: (1^(2))/(1.3)+(2^(2))/(3....

Statement -1: `(1^(2))/(1.3)+(2^(2))/(3.5)+(3^(2))/(5.7)+ . . . .+(n^(2))/((2n-1)(2n+1))=(n(n+1))/(2(2n+1))` Statement -2: `(1)/(1.3)+(1)/(3.5)+(1)/(5.7)+ . . . .+(1)/((2n-1)(2n+1))=(1)/(2n+1)`

A

Statement -1 is true, Statement -2 is True, Statement -2 is a correct explanation for Statement for Statement -1.

B

Statement -1 is true, Statement -2 is True, Statement -2 is not a correct explanation for Statement for Statement -1.

C

Statement -1 is true, Statement -2 is False.

D

Statement -1 is False, Statement -2 is True.

Text Solution

Verified by Experts

The correct Answer is:
C
We, have
`(1^(2))/(1.3)+(2^(2))/(3.7)+(3^(2))/(5.7)+ . . . . . . . . . . . . . +(n^(2))/((2n-1)(2n+1))`
`=underset(r=1)overset(n)sum(r^(2))/((2r-1)(2r+1))`
`=(1)/(4)underset(r=1)overset(n)sum(4r^(2))/((2r-1)(2r+1))`
`=(1)/(4)underset(r=1)overset(n)sum((2r-1)(2r+1)+1)/((2r-1)(2r+1))`
`=(1)/(4)underset(r=1)overset(n)sum{1+(1)/((2r-1)(2r+1))}`
`=(1)/(4)underset(r=1)overset(n)sum1+(1)/(8)underset(r=1)overset(n)sum((1)/(2r-1)-(1)/(2r+1))`
`=(n)/(4)+(1)/(8)(1-(1)/(2n+1))=(n(n+1))/(2(2n+1))`
So, statement -1 is true.
Statement -2 is fals, because
`(1)/(1.3)+(1)/(3.5)+(1)/(5.7)+ . . . . .+(1)/((2n-1)(2n+1))`
`=underset(r=1)overset(n)sum(1)/((2r-1)(2r+1))=(1)/(2)underset(r=1)overset(n)sum((1)/(2r-1)-(1)/(2r+1))`
`=(1)/(2)(1-(1)/(2n+1))=(n)/(2n+1)`
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