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 Figure). In how many rows are the 200 logs placed and how many logs are in the top row?

Given:

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.

To do:

We have to find the number of rows in which the 200 logs placed and the number of logs in the top row.

Solution:

Let the number of rows be $n$

The number of logs in the top be $a$

Here,

$a_{n}=20, d=1$ and $S_{n}=200$

We know that,

$a_{n}=a+(n-1) d$

$20=a+(n-1) 1$

$a=20-n+1$

$a=21-n$

$400=n[a+20]$

$n a+20 n=400$

$n(21-n)+20 n=400$

$21 n-n^{2}+20 n=400$

$n^{2}-41 n+400=0$

$n^{2}-25 n-16 n+400=0$

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

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

$n-16=0$ or $n-25=0$

$n=16$ or $n=25$

If $n=16$, then,

$a+(n-1) d=20$

$a+(16-1) d=20$

$a+15 d=20$

$a+15(1)=20$

$a=20-15=5$

If $n=25$, then,

$a+(25-1) d=20$

$a+24(1)=20$

$a=20-24=-4$ which is not possible.

Hence, the number of rows are $16$ and the number of logs in the top row is $5$.

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Updated on: 10-Oct-2022

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