Question

Which equation could be used to calculate the sum of the geometric series?

1/3 + 2/9 + 4/27 + 8/81 + 16/243

Answer 1

A

Explanation:

got it right on edge

Answer 2

The equation is:

`S=(a_n*r-a_1)/(r-1)`

`S=((16)/(243)*(2)/(3)-(1)/(3))/((2)/(3)-1)`

The sum is:

`S=(211)/(243)`

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If the sequence is infinite, the formula is:

`S = (a_1)/(1-r)`

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Explanation:

We must calculate the radius of the geometric series

`r =(a_(n+1))/(a_n)\\\\r=((2)/(9))/((1)/(3))\\\\r=(2)/(3)`

The first term of the series is:
`a_1=(1)/(3)`

The last term of the series is:
`a_n=(16)/(243)`

If the sequence is finite then the formula is:

`S=(a_n*r-a_1)/(r-1)`

`S=((16)/(243)*(2)/(3)-(1)/(3))/((2)/(3)-1)`

`S=(211)/(243)`

If the sequence is infinite then by definition as the radius are
`0 <| r | <1` then the formula for the sum of the geometric sequence is:

`S = (a_1)/(1-r)\\\\S = ((1)/(3))/(1-(2)/(3))\\\\S =1`