In Problem (b), we are dealing with a geometric series. The formula for the sum of the first $n$ terms of a geometric series is:

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

where:

• $a$ is the first term of the series,
• $r$ is the common ratio,
• $n$ is the number of terms.

For the series $1 + 5 + 5^2 + \cdots + 5^{n-1}$:

• The first term $a = 1$,
• The common ratio $r = 5$,
• The number of terms is $n$.

Applying the formula:

$S_n = 1 \cdot \frac{5^n - 1}{5 - 1} = \frac{5^n - 1}{4}$

This is how we arrive at the formula given in Problem (b):

$1 + 5 + 5^2 + \cdots + 5^{n-1} = \frac{1}{4}(5^n - 1)$

Now, regarding the $4.5$:

It seems there is some misunderstanding in the notation or interpretation of the series sum. There is no specific value of 4.5 mentioned directly in the sum formula or the general series result. If you meant the constant divisor 4 in the formula $\frac{1}{4}(5^n - 1)$, this comes directly from the geometric series sum formula where the denominator is $r - 1 = 5 - 1 = 4$.

If there's a specific step or value calculation you're referring to, please clarify, and I'd be happy to help explain further.

Would you like further clarification on any part of this?

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Related Questions:

1. How is the formula for the sum of a geometric series derived?
2. Why is the common ratio in this series considered to be 5?
3. What happens to the series sum if the common ratio $r$ is less than 1?
4. Can you generalize the formula for a geometric series where the first term isn't 1?
5. How does the sum of a finite geometric series differ from an infinite geometric series?
6. What are some common applications of geometric series in real life?
7. How would the sum formula change if the series was decreasing instead of increasing?
8. Can the sum formula be used for non-integer values of $n$?

Tip:

When working with geometric series, always identify the first term and common ratio correctly, as these are critical to applying the formula for the series sum.