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?

The number of logs in each row forms an Arithmetic Progression (A.P.):

20, 19, 18,……

Total sum of logs = 200.

S_n = 200

Common difference (d) = 19 - 20 = -1

We know that,

Sum of n terms of an A.P. is given by,

∴ S_n = $\dfrac{n}{2}$ [2a + (n - 1)d]

⇒ 200 = $\dfrac{n}{2}$ [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.