Let's compute each sum step by step.

First Sum:

$5 + 5(3) + 5(3)^2 + 5(3)^3 + ... + 5(3)^7$ This is a geometric series with:

• First term: $a = 5$
• Common ratio: $r = 3$
• Number of terms: $n = 8$ (since it starts with $5(3)^0$)

The sum of a geometric series is given by:

$S_n = a \frac{r^n - 1}{r - 1}$

Substituting the values:

$S_8 = 5 \frac{3^8 - 1}{3 - 1}$

Calculating:

$3^8 = 6561$

$S_8 = 5 \times \frac{6561 - 1}{2} = 5 \times \frac{6560}{2} = 5 \times 3280 = 16400$

Final Answer:
$\mathbf{16,400}$

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Second Sum:

$\sum_{j=1}^{9} \left(-\frac{1}{2}\right)^j$

This is also a geometric series with:

• First term: $a = -\frac{1}{2}$
• Common ratio: $r = -\frac{1}{2}$
• Number of terms: $n = 9$

Using the sum formula:

$S_n = a \frac{1 - r^n}{1 - r}$

Substituting the values:

$S_9 = -\frac{1}{2} \times \frac{1 - (-\frac{1}{2})^9}{1 - (-\frac{1}{2})}$

Calculating:

$(-\frac{1}{2})^9 = -\frac{1}{512}$

$1 - (-\frac{1}{512}) = 1 + \frac{1}{512} = \frac{513}{512}$

$1 - (-\frac{1}{2}) = 1 + \frac{1}{2} = \frac{3}{2}$

Now computing the fraction:

$S_9 = -\frac{1}{2} \times \frac{513}{512} \times \frac{2}{3}$

$S_9 = -\frac{513}{1536}$

Simplifying:

$S_9 = -\frac{171}{512}$

Final Answer:
$\mathbf{-\frac{171}{512}}$

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Would you like further explanation or a breakdown of any steps?

Related Questions:

1. How do you determine the sum of an infinite geometric series?
2. What is the general formula for the sum of a geometric series?
3. How do you determine the convergence of a geometric series?
4. Can a geometric series have a negative common ratio? What happens in that case?
5. How do you convert a repeating decimal into a geometric series?

Tip:

When working with geometric series, always check whether the series is finite or infinite, as the sum formula differs in each case!