Question

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

Answer 1

No. of logs in bottom row $=20$

No. of logs in next row $=19$

No. of logs in next row to it $=18$

Hence the AP series is

$20,19,\mathrm{18.......}$

Where first term $\left(a\right)=20$

and common difference $\left(d\right)=-1$

$\therefore S_n=\frac{n}{2}[2a+(n-1\left)d\right]$ Let n rows are required to the $200$ logs]

$\therefore \frac{n}{2}[2a+(n-1\left)d\right]=200$

$\Rightarrow \frac{n}{2}[2\times 20+(n-1\left)\right(-1\left)\right]=200$

$\Rightarrow 40n=n^2+n=400$

$\Rightarrow n^2-4n+400=0$

$\Rightarrow n^2-25n-16n+400=0$

$\Rightarrow n(n-25)-16(n-25)=0$

$\Rightarrow (n-16)(n-25)=0$

$\therefore n=16,25$

As $25$ rows are not popssible

$\therefore$ No. of rows required $=16$

And , no. of logs placed in top (16th) row $=5$