200 logs are stacked so that there are 20 logs in the bottom row, 19 logs in the next row, 18 in the next, and so on. How many rows are formed and how many logs are there in the top row?
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200 logs are stacked so that there are 20 logs in the bottom row, 19 logs in the next row, 18 in the next, and so on. How many rows are formed and how many logs are there in the top row?

KnowledgeBoat · Accepted Answer

The number of logs in each row forms an Arithmetic Progression (A.P.): 20, 19, 18,...... Total sum of logs = 200. Sn = 200 Common difference (d) = 19 - 20 = -1 We know that, Sum of n terms of an A.P. is given by, ∴ Sn = [2a + (n - 1)d] ⇒ 200 = [2(20) + (n - 1)(-1)] ⇒ 200 × 2 = n[40 - n + 1] ⇒ 400 = n(41 - n) ⇒ 400 = 41n - n2 ⇒ n2 - 41n + 400 = 0 ⇒ n2 - 16n - 25n + 400 = 0 ⇒ n(n - 16) - 25(n - 16) = 0 ⇒ (n - 25)(n - 16) = 0 ⇒ (n - 25) = 0 or (n - 16) = 0 ⇒ n = 16 or n = 25. Case 1 : If n = 25, logs in the 25th row a25: ⇒ a25 = a + (25 - 1)d = 20 + (24)(-1) = 20 - 24 = -4 (the number of logs cannot be negative). Thus, n 25. If n = 16 logs in the 16th row a16: ⇒ a16 = a + (16 - 1)d = 20 + (15)(-1) = 20 - 15 = 5. Hence, no. of rows = 16 and the top row has 5 logs.

Mathematics

200 logs are stacked so that there are 20 logs in the bottom row, 19 logs in the next row, 18 in the next, and so on. How many rows are formed and how many logs are there in the top row?

AP

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200 logs are stacked so that there are 20 logs in the bottom row, 19 logs in the next row, 18 in the next, and so on. How many rows are formed and how many logs are there in the top row?

AP

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