Solutions - 0150-02

(a)

The first term is $a_0=5^{-0}=1$. The common ratio is $1/5$. Therefore the sum:

$$S=\frac{a_0}{1-r}\gg \gg \frac{1}{1-\frac{1}{5}}\gg \gg \frac{5}{4}$$

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(b)

Split numerator and obtain two geometric series:

$$\sum _{n=0}^{\infty }\frac{2+3^n}{5^n}\gg \gg \sum _{n=0}^{\infty }2{\left(\frac{1}{5}\right)}^n+\sum _{n=0}^{\infty }{\left(\frac{3}{5}\right)}^n$$

Geometric series total sum formula:

$$\gg \gg \frac{2}{1-\frac{1}{5}}+\frac{1}{1-\frac{3}{5}}\gg \gg 5$$