To create an explicit formula for the number of bricks in each row of the pile described, we can observe the pattern of the number of bricks in each row.
- The first row has 97 bricks.
- The second row has 91 bricks.
- The third row has 85 bricks.
- The fourth row has 79 bricks.
- This pattern continues until we reach the top row with 1 brick.
We notice that with each row, the number of bricks decreases by 6. Thus, we have:
- Row 1: \(97\)
- Row 2: \(91\)
- Row 3: \(85\)
- Row 4: \(79\)
- ...
- Row \(n\): \(97 - 6(n - 1)\)
Where \(n\) represents the row number, starting from 1.
Explicit Formula
The explicit formula for the number of bricks in row \(n\) is:
\[ B(n) = 97 - 6(n - 1) \]
This simplifies to:
\[ B(n) = 97 - 6n + 6 = 103 - 6n \]
Finding the Number of Rows
To find out how many rows exist until we reach 1 brick, we set \(B(n) = 1\):
\[ 103 - 6n = 1 \]
Solving for \(n\):
\[ 103 - 1 = 6n \ 102 = 6n \ n = \frac{102}{6} = 17 \]
Therefore, there are 17 rows in total.
Conclusion
The explicit formula for the number of bricks in the \(n\)-th row is:
\[ B(n) = 103 - 6n \]
And there are a total of 17 rows in this pile of bricks.