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

C

Explanation:

Answer 2

`(1)/(3)+(2)/(9)+(4)/(27)+(8)/(81)+(16)/(243) = \\ \\\\ = \sum\limits_(k=1)^(5)(2^(k-1))/(3^k) = \sum\limits_(k=1)^(5)(2^(k))/(2\cdot 3^k) = (1)/(2)\cdot \sum\limits_(k=1)^(5)(2^(k))/(3^k) = \\ \\\\ = (1)/(2)\cdot \sum\limits_(k=1)^(5)\Big((2)/(3)\Big)^k = (1)/(2)\cdot \left[\Big((2)/(3)\Big)^1+\Big((2)/(3)\Big)^2+...+\Big((2)/(3)\Big)^5\right] =`

`= (1)/(2)\cdot ((2)/(3)\cdot\left[\Big((2)/(3)\Big)^5-1\right])/((2)/(3)-1) =-\Big((2)/(3)\Big)^(5)+1 = (-2^5+3^5)/(3^5) = \boxed{(211)/(243)}`

`\text{I used the geometric series formula for sum: }S_n = (b_1\cdot (q^n - 1))/(q-1)`