Question

If `(48)/(2.3)+(47)/(3.4)+(46)/(4.5)+ . . . +(2)/(48.29)+(1)/(49.50)`
`=(51)/(2)+k(1+(1)/(2)+(1)/(3)+ . . .+(1)/(50))`, then k equals
A. 2
B. `-1`
C. `-(1)/(2)`
D. 1

Answer 1

Correct Answer - B
We have,
`(48)/(2.3)+(47)/(3.4)+(46)/(4.5)+ . . . +(2)/(48.29)+(1)/(49.50)`
`=(51)/(2)+k(1+(1)/(2)+(1)/(3)+ . . .+(1)/(50))`
`rArr""underset(r=2)overset(49)sum(50-r)/(r(r+1))=(51)/(2)+(1+(1)/(2)+(1)/(3)+ . .. +(1)/(50))`
`rArr" "50underset(r=2)overset(49)sum(1)/(r(r+1))-underset(r=2)overset(49)sum(1)/(r+1)=(51)/(2)+k(1+(1)/(2)+(1)/(3)+ . . . +(1)/(50))`
`rArr" "50underset(r=2)overset(49)sum((1)/(r)-(1)/(r+1))-underset(r=2)overset(49)sum(1)/(r+1)=(51)/(2)+k(1+(1)/(2)+(1)/(3)+ . . . .+(1)/(50))`
`rArr" "50((1)/(2)-(1)/(50))-underset(r=2)overset(49)sum(1)/(r+1)=(51)/(2)+k(1+(1)/(2)+(1)/(3)+ . . . +(1)/(50))`
`rArr" "24-((1)/(3)+(1)/(4)+ . . .+(1)/(50))=(51)/(2)+k(1+(1)/(2)+(1)/(3)+ . . .. +(1)/(50))`
`rArr" "24-(1+(1)/(2)+(1)/(3)+ . . . +(1)/(50))+1+(1)/(2)=(51)/(2)+k(1+(1)/(2)+ . . .+(1)/(50))`
`rArr" "(51)/(2)-(1+(1)/(2)+(1)/(3)+ . . . +(1)/(50))=(51)/(2)+k(1+(1)/(2)+ . .. . +(1)/(50))`
`rArr" "-(1+(1)/(2)+(1)/(3)+ . . . +(1)/(50))=k(1+(1)/(2)+ . . .. +(1)/(50))`
`rArr" "k=-1`