Q : 19      \small 200  logs are stacked in the following manner: \small 20 logs in the bottom row, \small 19  in the next row, \small 18 in the row next to it and so on (see Fig. \small 5.5). In how many rows are the \small 200 logs placed and how many logs are in the top row?

               

Answers (1)
G Gautam harsolia

As, the rows are going up, the no of logs are decreasing, 
We can clearly see that 20, 19, 18, ..., is an AP .
and here  a = 20 \ and \ d = -1 
Let suppose 200 logs are arranged in 'n' rows,
Then, 
S_n = \frac{n}{2}\left \{ 2\times 20 +(n-1)(-1) \right \}
200 = \frac{n}{2}\left \{ 41-n \right \}
\Rightarrow n^2-41n +400 = 0
\Rightarrow n^2-16n-25n +400 = 0
\Rightarrow (n-16)(n-25) = 0
\Rightarrow n = 16 \ \ and \ \ n = 25
Now,
case (i) n = 25
a_{25} =a+24d = 20+24\times (-1)= 20-24=-4
But number of rows can not be in negative numbers 
Therefore, we will reject the value n = 25

case (ii) n = 16

a_{16} =a+15d = 20+15\times (-1)= 20-15=5
Therefore, the number of rows in which 200 logs are arranged is equal to 5


 

 

 

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