ANSWER
[tex]S_ \infty = \frac{40}{3} [/tex]
EXPLANATION
The given series is
[tex]20-10+5-...[/tex]
The first term of the series is,
[tex]a_1=20[/tex]
The common ratio of this series is,
[tex]r = \frac{ - 10}{20} [/tex]
This simplifies to,
[tex]r = - \frac{1}{2} [/tex]
The sum to infinity of this sequence is given by the formula,
[tex]S_ \infty = \frac{a_1}{1-r} [/tex]
We substitute the above values into the formula to obtain,
[tex]S_ \infty = \frac{20}{1- - \frac{1}{2} } [/tex]
This simplifies to,
[tex]S_ \infty = \frac{20}{1 + \frac{1}{2} } [/tex]
We simplify the denominator to get,
[tex]S_ \infty = \frac{20}{ \frac{3}{2} } [/tex]
This will finally give us,
[tex]S_ \infty =20 \times \frac{2}{3} [/tex]
[tex]S_ \infty = \frac{40}{3} [/tex]
The correct answer is A.
