Question

NO LINKS!! Find the sum of the infinite geometric series:

-125/36 + 25/6 - 5 + 6 - . . .

Answer 1

`S_(\infty)=-(625)/(396)`

Explanation:

`\boxed{\begin{minipage}{5.5 cm}\underline{Sum of an infinite geometric series}\\\\$S_(\infty)=(a)/(1-r)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}`

Given geometric series:

`-(125)/(36)+(25)/(6)-5+6-...`

To find the common ratio, divide a term by the previous term:

`\implies r=(a_4)/(a_3)=-(6)/(5)`

Substitute the found common ratio and given first term into the sum formula:

`\implies S_(\infty)=(-(125)/(36))/(1-\left(-(6)/(5)\right))`

`\implies S_(\infty)=(-(125)/(36))/(1+(6)/(5))`

`\implies S_(\infty)=(-(125)/(36))/((11)/(5))`

`\implies S_(\infty)=-(125)/(36) *(5)/(11)`

`\implies S_(\infty)=-(625)/(396)`