The sum to infinity of the series `1+2/3+6/(3^2)+(10)/(3^3)+(14)/(3^4). . . . . .` is (1) 2 (2) 3 (3) 4 (4) 6

Options

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

Answer

A

Explanation

Let S`=1+(2)/(3)+(6)/(3^(2))+(10)/(3^(3))+(14)/(3^(4))+ . . .`
`impliesS-1=(2)/(3)+(6)/(3^(2))+(10)/(3^(3))+(14)/(3^(4))+` . . . (i)
`implies(S-1)/(3)=(2)/(3^(2))+(6)/(3^(3))+(10)^(3^(4))+(14)/(3^(5))+` . . . (ii)
On subtracting Eq. (ii) from Eq. (i) we get
`(2)/(3)(S-1)=(2)/(3)+(4)/(3^(2))+(4)/(3^(3))+(4)/(3^(4))+` . . .
`impliesS-1=1+(2)/(3)+(2)/(3^(2))+(2)/(3^(3))` + . .
`impliesS=2+((2)/(3))/(1-(1)/(3))`
`=2+1=3`