Find the sum of the following series to `n` terms `5+7+13+31+85+`

Options

A. `4n+(1)/(2)(3^(n)-1)`
B. `8n+(1)/(2)(3^(n)-1)`
C. `2n+(1)/(2)(3^(n)-1)`
D. none of these

Answer

A

Explanation

We observe that the sequence of the differences between the successive terms is
2,6,18,54, . . . . . . . .
Clearly, it is a G.P. with common ratio 3. So, let the `n^(th)` term of the given series be
`T_(n)=a.3^(n-1)+bn+c`
Putting n=1,2,3, we get
`T_(1)=a+b+crArra+b+c=5`
`T_(2)=3a+2b+crArr3a+2b+c=7`
`T_(3)=9a+3b+crArr9a+3b+c=13`
Solving these equations, we get
a=1,b=0,c=4
Putting the values of a,b,c, in (i), we get
`T_(n)=3^(n-1)+4`
So, sum of the series is given by
`underset(r=1)overset(n)sumT_(r)=underset(r=1)overset(n)sum(3^(r-1)+4)`
`rArr" "underset(r=1)overset(n)sumT_(r)=((3^(n)-1)/(3-1))+4n=(3^(n)-1)/(2)+4n=4n+(1)/(2)(3^(n)-1)`