Question

200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the next row to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row?

Answer 1

Hint: According to the question, we have to first figure out what we need to find. We can see that the series is an AP series. So, we can apply the formula for the AP series, and solve the ‘n’ part to get the answer.

Formula used:
 \[{S_n} = \dfrac{n}{2}(2a + (n - 1)d)\]

Complete step by step solution:
First, we need to assign a value to the number of logs on each row. So, let the number of logs on the first row be \[{{x_1}}\] . According to the question, we know that the first row had 20 logs, so we get:
 \[{{x_1 = 20logs}}\]
Similarly, we will assume that the number of logs on the second row will be \[{{x_2}}\] . From the question, we know that the second row had 19 logs, so we get:
 \[{{x_2 = 19logs}}\]
Now the difference between the number of logs in both the rows is:
 \[difference = x_2 - x_1\]
 \[ \Rightarrow x_2 - x_1 = 19 - 20\]
 \[ \Rightarrow difference = - 1\]
Now, for the third row, \[{{x_3 = 18logs}}\]
The difference is:
 \[ \Rightarrow x_3 - x_2 = 18 - 19\]
 \[ \Rightarrow difference = - 1\]
Now, we can see that the series is \[20,\,19,\,18,\,......\] and so on.
The difference is the same and constant. So, we can say that it is an AP series.
We need to know the total number of rows where 200 logs can fit. So, we know that:
 \[{S_n} = 200\]
We have to find the ‘n’.
We will try to calculate it with the help of the formula:
 \[{S_n} = \dfrac{n}{2}(2a + (n - 1)d)\]
Here, ‘a’ is the number of logs for the first row, ‘d’ is the difference.
 \[ \Rightarrow 200 = \dfrac{n}{2}(2a + (n - 1)d)\]
 \[ \Rightarrow 200 = \dfrac{n}{2}(2 \times 20 + (n - 1) - 1)\]
 \[ \Rightarrow 200 = \dfrac{n}{2}(40 + n( - 1)( - 1)( - 1))\]
 \[ \Rightarrow 200 = \dfrac{n}{2}(40 - n + 1)\]
 \[ \Rightarrow {n^2} - 41n + 400 = 0\]
Now, we will do factorisation by splitting the middle term, and we get:
 \[ \Rightarrow {n^2} - 16n - 25n + 400 = 0\]
 \[ \Rightarrow n(n - 16) - 25(n - 16) = 0\]
 \[ \Rightarrow (n - 25)(n - 16) = 0\]
 \[ \Rightarrow n = 25\,or\,n = 16\]
When ‘n’ is becoming 16, then:
 \[n\_16 = 21 - 16\]
 \[n\_16 = 5\]
When ‘n’ is becoming 25, then:
 \[n\_25 = 21 - 25\]
 \[ \Rightarrow n\_25 = - 4\]
The number of logs can never be negative, so we will consider ‘n’ to be 16.
Therefore, there are 16 rows and the top row has 5 logs.
So, the correct answer is “16”.

Note: The full form for AP is Arithmetic Progression. It is a series of numbers in which there is an order where when there is any difference between two consecutive numbers, then the difference is a constant value.