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Let's break down and solve the given problem.

The problem describes an arithmetic sequence where:
- On the first day, Gloria plants 5 flowers.
- The number of flowers planted each day forms an arithmetic sequence.
- On the last day of September (the 30th day), she plants 63 flowers.

### Part (a): Find the common difference of the sequence.

In an arithmetic sequence, the number of flowers planted on the \(n\)-th day is given by:

\[
a_n = a_1 + (n - 1) \cdot d
\]

where:
- \(a_n\) is the number of flowers on the \(n\)-th day,
- \(a_1 = 5\) is the number of flowers planted on the first day,
- \(d\) is the common difference we need to find,
- \(n = 30\) and \(a_{30} = 63\) are the conditions for the last day.

Using this, we have the equation for the 30th day:

\[
63 = 5 + (30 - 1) \cdot d
\]

Simplifying:

\[
63 = 5 + 29d
\]
\[
63 - 5 = 29d
\]
\[
58 = 29d
\]
\[
d = \frac{58}{29} = 2
\]

So, the common difference \(d\) is **2**.

---

### Part (b): Find the total number of flowers Gloria is going to plant during September.

The sum of the first \(n\) terms of an arithmetic sequence is given by:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]

Here:
- \(n = 30\),
- \(a_1 = 5\),
- \(a_{30} = 63\).

Substituting into the formula:

\[
S_{30} = \frac{30}{2} \cdot (5 + 63)
\]
\[
S_{30} = 15 \cdot 68 = 1020
\]

So, the total number of flowers Gloria will plant during September is **1020**.

---

### Part (c): Calculate the percentage error in Gloria’s estimate.

Gloria estimated that she would plant 1000 flowers, but in reality, she will plant 1020. The percentage error is calculated as:

\[
\text{Percentage error} = \frac{|\text{Actual} - \text{Estimate}|}{\text{Actual}} \times 100
\]

Substituting the values:

\[
\text{Percentage error} = \frac{|1020 - 1000|}{1020} \times 100 = \frac{20}{1020} \times 100 \approx 1.96\%
\]

So, the percentage error in Gloria’s estimate is approximately **1.96%**.

---

Would you like further explanation on any part of the solution, or to go deeper into related concepts?

### Related Questions:
1. What are the general properties of an arithmetic sequence?
2. How would the sum formula change if the sequence was geometric instead of arithmetic?
3. How does the common difference affect the shape of an arithmetic sequence's graph?
4. Can you derive the common difference if only non-consecutive terms are given?
5. What is the relationship between the average number of flowers planted and the total number of flowers?

#### Tip:
When solving arithmetic sequences, always write down the known variables first. This helps to set up the equation correctly for finding unknowns like the common difference or sum.

On the first day of September, 2019, Gloria planted 5 flowers in her garden. The number of flowers she plants each day forms an arithmetic sequence. The number of flowers she is going to plant on the last day of September is 63.

(a) Find the common difference of the sequence.
(b) Find the total number of flowers Gloria is going to plant during September.
(c) Gloria estimated she would plant 1000 flowers in the month of September. Calculate the percentage error in Gloria's estimate.

Solve an arithmetic sequence problem where Gloria plants 5 flowers on the first day of September and 63 flowers on the last day. Find the common difference, the total number of flowers planted, and the percentage error in Gloria's estimate. Suitable for Grades 8-10 students.

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