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

Question

Filo · Accepted Answer

It can be observed that the numbers of logs in rows are in an A.P.

20, 19, 18 \dots

For this A.P.,

a = 20

,

d = 19 - 20 = - 1

Let a total of

200

logs be placed in n rows.

S_{n} = 200

S_{n} = \frac{n}{2} (a + (n - 1) d

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

\Rightarrow 400 = n (40 - n + 1)

\Rightarrow 400 = n (41 - n)

\Rightarrow 400 = 41 n - n^{2}

\Rightarrow n^{2} - 16 n - 25 n + 400

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

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

Either

(n - 16) = 0

or

(n - 25) = 0

n = 16

or

n = 25

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

\Rightarrow a_{16}

= 20 + (16 - 1) (- 1)

\Rightarrow a_{16}

20 - 15

a_{16}

= 5

Similarly,

a_{25}

= 20 + (25 - 1) (- 1)

a_{25}

= 20 - 24

= - 4

Clearly, the number of logs in

1 6^{t h}

row is

5

. However, the number of logs in

2 5^{t h}

row is negative, which is not possible.

Therefore,

200

logs can be placed in

16

rows and the number of logs in the

1 6^{t}

h row is

5

.

Question Text	$200$ 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. In how many rows are the $200$ logs placed and how many logs are in the top row.
Answer Type	Text solution:1
Upvotes	150

$200$ 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. In how many rows are the $200$ logs placed and how many logs are in the top row.

Text solutionVerified

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