The problem in the image describes a decreasing number of blocks per row, starting with 87 blocks in the bottom row, and reducing the number of blocks by 6 in each successive row, with the top row containing 3 blocks.

This forms an arithmetic sequence where:

• First term ($a_1$) = 87
• Common difference ($d$) = -6
• Last term ($a_n$) = 3

We can solve the following two questions:

1. How many rows are there in the pile?
2. What is the total number of blocks in the pile?

Let me solve these step-by-step:

1. Number of Rows:

In an arithmetic sequence, the general term formula is: $a_n = a_1 + (n-1) \cdot d$ We know $a_1 = 87$, $d = -6$, and $a_n = 3$. Plugging these into the formula: $3 = 87 + (n-1) \cdot (-6)$ Simplifying this equation: $3 = 87 - 6(n-1)$ $3 = 87 - 6n + 6$ $3 = 93 - 6n$ $6n = 93 - 3$ $6n = 90$ $n = \frac{90}{6} = 15$ So, there are 15 rows in total.

2. Total Number of Blocks:

The sum of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (a_1 + a_n)$ Substituting the known values: $S_{15} = \frac{15}{2} \cdot (87 + 3)$ $S_{15} = \frac{15}{2} \cdot 90$ $S_{15} = 15 \cdot 45 = 675$ So, the total number of blocks is 675.

Would you like further details or have any questions? Here are 5 related questions to think about:

1. What happens if the common difference was different, say -4 instead of -6?
2. How would the sum change if the top row had more than 3 blocks, like 5 blocks?
3. Can you derive a formula for the number of rows based on the common difference and the top row value?
4. What real-world structures could use this type of block arrangement?
5. How could you find the average number of blocks per row?

Tip: For arithmetic sequences, always check if the pattern is consistent and use the general term formula to make predictions.