250 logs are stacked in the following manner : 22 logs in the bottom row, 21 in the next row, 20 in the row next to it and so on (as shown by an example). In how many rows, are the 250 logs placed and how many logs are there in the top row ?
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding rag
Model Answer
The number of logs in each row forms an AP:
- First term $a = 22$, common difference $d = -1$, Total $S_n = 250$
Finding number of rows (n):
Using $S_n = \dfrac{n}{2}[2a + (n-1)d]$
$$250 = \frac{n}{2}[2(22) + (n-1)(-1)]$$
$$500 = n[44 - n + 1] = n[45 - n]$$
$$500 = 45n - n^2$$
$$n^2 - 45n + 500 = 0$$
$$(n - 20)(n - 25) = 0$$
So $n = 20$ or $n = 25$.
Finding the top row for each value of n:
If $n = 25$: $a_{25} = 22 + (25-1)(-1) = 22 - 24 = -2$
Since number of logs cannot be negative, $n = 25$ is rejected.
If $n = 20$: $a_{20} = 22 + (20-1)(-1) = 22 - 19 = 3$
∴ The 250 logs are placed in 20 rows and the top row has 3 logs.
Source: Chapter 5, Section 5.4
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Explanation
- Identify this as an AP sum problem: rows decrease by 1 log each time, so $d = -1$.
- Set up $S_n = 250$ and solve the quadratic — you get two values of $n$.
- Always check both values: $n = 25$ gives a negative number of logs in the top row, which is impossible, so reject it.
- Examiners award marks for: correct AP identification, correct formula use, solving the quadratic, rejecting $n = 25$ with reason, and stating the final answer clearly.