Question

Which equation could be used to calculate the sum of the geometric series? 1/4+2/9+4/27+8/81+16/243?

Answer 1

Answer: Sum of the geometric series will be
`(763)/(972)`

Explanation:

Since we have given that

`(1)/(4)+(2)/(9)+(4)/(27)+(8)/(81)+(16)/(243)`

Here,

`a=(2)/(9)\\\\r=(a_2)/(a_1)\\\\r=((4)/(27))/((2)/(9))=(4)/(27)* (9)/(2)=(2)/(3)\\\\n=4`

As we know the formula for "Sum of n terms in geometric series ":

`S_n=(a(1-r^n))/(1-r)\\S_n=((2)/(9)(1-(2)/(3)^4))/(1-(2)/(3))\\S_n=(130)/(243)`

So, Complete sum will be

`(130)/(243)+(1)/(4)=(520+243)/(972)=(763)/(972)`

Hence, Sum of the geometric series will be
`(763)/(972)`

Answer 2

From the given sequence:
1/4, 2/9, 4,27, 8/81, 16/243
the common ratio is: 2/3
thus the sum of the series will be given by the formula:
Sn=[a(1-r^n)]/(1-r)
plugging the values we obtain:

Sn=[1/4(1-(2/3)^n)]/(1-2/3)
thus the equation that will be used to find the sum is:
Sn=3[1/4-1/4(2/3)^n]
=3/4[1-(2/3)^n]