If the sum of the first terms of the series
`(1(3)/5)^2+(2(2)/(5))+(3(1)/(5))^2+4^2+(4(4)/(5))+..... is 16/5 m `
then m is equal to

Options

A. 102
B. 101
C. 100
D. 99

Answer

B

Explanation

Let S be the sum of the following series.
`(1(3)/(5))^(2)+(2(2)/(5))^(2)+(3(1)/(5))^(2)+4^(2)+ . . . . .` upto 10 terms
i.e. `S=((8)/(5))^(2)+((12)/(5))^(2)+((16)/(5))^(2)+((20)/(5))^(2)+((24)/(5))^(2)+ . . . . . .` upto 10 terms
`rArr" "S=underset(r=1)overset(10)sum{(8+4(r-1))/(5)}^(2)`
`rArr" "S=(16)/(25)underset(r=1)overset(10)sum(r+1)^(2)`
`rArr" "S=(16)/(25){underset(r=0)overset(10)sum(r+1)^(2)-1^(2)}`
`rArr" "(16)/(25)m=(16)/(25){(11xx12xx23)/(6)-1}" "{:[becauseS=(16)/(5)m"(given)"]:}`
`rArr" "m=(1)/(5)(22xx23-1)rArrm=101`