The problem presented in the image can be summarized as follows:

Raymond numbered some gifts and arranged them in 36 rows. The first row contains 8 gifts, and each subsequent row contains 7 more gifts than the previous one.

Part (a): Find the number of gifts in the last row.
Part (b): Find the total number of gifts in the first $n$ rows. Also, determine the row in which gift number 250 is located.

Let's solve these step-by-step.

Part (a): Find the number of gifts in the last row.

This situation can be modeled using an arithmetic sequence, where:

• The first term $a_1 = 8$,
• The common difference $d = 7$,
• The total number of rows $n = 36$.

The general formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$

Plugging in the known values: $a_{36} = 8 + (36 - 1) \cdot 7 = 8 + 35 \cdot 7 = 8 + 245 = 253$

So, the number of gifts in the last row (the 36th row) is 253 gifts.

Part (b): Find the total number of gifts in the first $n$ rows.

The sum of the first $n$ terms of an arithmetic sequence can be calculated using the formula: $S_n = \frac{n}{2} \cdot (a_1 + a_n)$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of rows.

For 36 rows: $S_{36} = \frac{36}{2} \cdot (8 + 253) = 18 \cdot 261 = 4698$

Thus, the total number of gifts in the first 36 rows is 4698 gifts.

To determine the row in which gift number 250 is located:

We need to sum the number of gifts row by row until the sum reaches or exceeds 250. The number of gifts in the first row is 8, the second row is 15, the third row is 22, and so on. These values correspond to the terms of the arithmetic sequence.

To find this efficiently, let's find the smallest row $r$ such that the total number of gifts up to that row is greater than or equal to 250. I'll compute that now.The 250th gift is located in row 10. After summing the gifts in the first 9 rows, the total is less than 250, but by row 10, the total reaches 260, meaning the 250th gift falls within this row.

Would you like more details on any part of the solution? Here are some related questions to further explore:

1. How do arithmetic sequences model real-world problems like this one?
2. How would the solution change if the common difference were something other than 7?
3. What would be the formula for the total number of gifts if we only had $n$ rows instead of 36?
4. Can you determine the gift number that appears in row 20?
5. How can we determine the row for a specific gift number more quickly using formulas?

Tip: In arithmetic sequences, the sum and term formulas are powerful tools for analyzing patterns in real-world problems like this.