Let’s analyze and solve the two sums in the uploaded image. Here are the steps for each one:

First Sum:

$5 + 5\left(-\frac{4}{9}\right) + 5\left(-\frac{4}{9}\right)^2 + \cdots$ This is an infinite geometric series where:

• First term ($a$) = $5$,
• Common ratio ($r$) = $-\frac{4}{9}$.

The sum of an infinite geometric series is given by: $S = \frac{a}{1 - r}, \quad \text{if } |r| < 1.$

Here: $S = \frac{5}{1 - \left(-\frac{4}{9}\right)} = \frac{5}{1 + \frac{4}{9}} = \frac{5}{\frac{9}{9} + \frac{4}{9}} = \frac{5}{\frac{13}{9}} = 5 \cdot \frac{9}{13} = \frac{45}{13}.$

Thus, the sum of the first series is: $\boxed{\frac{45}{13}}$

Second Sum:

$\sum_{i=1}^{\infty} 3\left(\frac{1}{5}\right)^i$ This is another infinite geometric series where:

• First term ($a$) = $3\left(\frac{1}{5}\right)$,
• Common ratio ($r$) = $\frac{1}{5}$.

The sum of this series is also given by: $S = \frac{a}{1 - r}, \quad \text{if } |r| < 1.$

Here: $S = \frac{3 \cdot \frac{1}{5}}{1 - \frac{1}{5}} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}.$

Thus, the sum of the second series is: $\boxed{\frac{3}{4}}$

Final Answers:

1. First sum: $\frac{45}{13}$,
2. Second sum: $\frac{3}{4}$.

Would you like a deeper explanation for either result?

Related Questions:

1. How does the formula for the sum of an infinite geometric series work?
2. What happens if $|r| \geq 1$ in a geometric series?
3. Can the sums of finite geometric series be calculated similarly?
4. What is the importance of convergence in infinite series?
5. How can these concepts be applied in real-world contexts?

Tip:

Always check if the absolute value of the common ratio ($|r|$) is less than 1 to determine if an infinite geometric series converges.