Question

200 logs are stacked in the following manner , 20 logs in the bottom row, 19 in the next row , 18 in the next to it and so on , In how may rows are the 200 logs placed and how many logs are in the top now ?​

Answer 1

$\\ \large\underline{ \underline{ \sf{ \red{given:} }}}$

• There are total 200 logs.

• 20 logs in bottom row , 19 in the next row , 18 in the next and do on.

$\\ \\ \large\underline{ \underline{ \sf{ \red{to \: find:} }}}$

• Total number of rows in which 200 logs are placed.

• Number of logs present in top row.

$\\ \\ \large\underline{ \underline{ \sf{ \red{solution:} }}}$

Total number of logs is 200.

$\\ \bigstar \boxed{ \bf \: s_{n} = \frac{n}{2}(2a + (n - 1)d }$

We know ,

• s(n) = 200

• a = 20

• d = 19-20 = -1

Putting values , we get ,

$\\ \sf \: 200 = \frac{n}{2} \{2(20) + (n - 1)( - 1) \} \\ \\ \sf \: 200 = \frac{n}{2} (40 - n + 1) \\ \\ \sf \: 200 \times 2 = n(41 - n) \\ \\ \sf \: 400 = 41n - {n}^{2} \\ \\ \sf \: {n}^{2} - 41n + 400 = 0 \\ \\ \sf \: {n}^{2} - 25n - 16n + 400 = 0 \\ \\ \sf \: n(n - 25) - 16(n - 25) = 0 \\ \\ \sf \: (n - 16)(n - 25) = 0 \\ \\ \sf \: (n - 16) = 0 \: \: \: \: \: or \: \: \: \: \: (n - 25) = 0 \\ \\ \sf \: n = 16 \: \: \: \: \: \: \: or \: \: \: \: \: \: n = 25$

Hence , number of rows are 16 or 25.

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We will find number of logs in both case.

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★ a(n) = a + (n-1)d

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For n = 16 ,

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→ a(16) = a + 15d

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→ a(16) = 20 + 15(-1)

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→ a(16) = 20-15

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→ a(16) = 5

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For n = 25 ,

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→ a(25) = a + (24d)

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→ a(25) = 20 + 24(-1)

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→ a(25) = 20-24

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→ a(25) = -4

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Negative value is not possible .

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Hence , Total number of rows is 16 and number of logs in top row is 5.