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Answer :
We can see that the numbers of logs in rows are in the form of an A.P.20, 19, 18…
For the given A.P.,
First term, a = 20
and common difference,d
\(= a_2 - a_1 = 19-20 = -1\)
Let a total of 200 logs be placed in n rows.
Thus, \(S_n = 200\)
By the sum of \(n^{th}\) term formula,
\(\Rightarrow S_n = {{n} \over {2}} [2a +(n -1)d]\)
\(\Rightarrow S_{12} = {{12} \over {2}} [2(20)+(n -1)(-1)]\)
\(\Rightarrow \) 400 = n (40- n+1)
\(\Rightarrow \) 400 = n (41-n)
\(\Rightarrow 400 = 41n-n^2\)
\(\Rightarrow n^2-41n + 400 = 0\)
\(\Rightarrow n^2-16n-25n+400 = 0\)
\(\Rightarrow \) n(n -16)-25(n -16) = 0
\(\Rightarrow\) (n -16)(n -25) = 0
Either (n -16) = 0 or n-25 = 0
\(\Rightarrow \) n = 16 or n = 25
By the nth term formula,
\(\Rightarrow a_n = a+(n-1)d\)
\(\Rightarrow a_{16} = 20+(16-1)(-1)\)
\(\Rightarrow a_{16} = 20-15\)
\(\Rightarrow a_{16} = 5\)
Similarly, the 25th term could be written as;
\(\Rightarrow a_{25} = 20+(25-1)(-1)\)
\(\Rightarrow a_{25} = 20-24\)
\(\Rightarrow a_{25}= -4 \)
It can be seen, the number of logs in 16th row is 5 as the numbers cannot be negative.
Therefore, 200 logs can be placed in 16 rows and the number of logs in the 16th row is 5.