The sum to infinity of the series `1+2/3+6/3^2+14/3^4+...is `

Options

A. 6
B. 2
C. 3
D. 4

Answer

C

Explanation

Let `S=1+(2)/(3)+(6)/(3^(2))+(10)/(3^(3))+(14)/(3^(4))+"......"" " "……..(i)`
`:. (1)/(3)S=(1)/(3)+(4)/(3^(2))+(6)/(3^(3))+(10)/(3^(4))+"......"" " "........(ii)"`
On subtracting Eq. (ii) from Eq.(i), we get
`(2)/(3)S=1+(1)/(3)+(4)/(3^(2))+(4)/(3^(3))+(4)/(3^(4))+"......"`
`=(4)/(3)+4((1)/(3^(2))+(1)/(3^(3))+(1)/(3^(4))+"......")=(4)/(3){((1)/(3^(2)))/(1-(1)/(3))}=(4)/(3)+(2)/(3)=2`
`:. S=3`.