Question

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 1

Given that,

The numbers of logs in rows are in the form of an A.P. 20, 19, 18…

From the given data

First-term, a = 20

Common difference, d = a2−a1 = 19−20 = −1

We have to find out in how many rows are the 200 logs placed and how many logs are in the top row

Let a total of 200 logs be placed in n rows.

Thus, S_n = 200

We know that the sum of nth term formula,

S_n = n/2 [2a +(n -1)d]

S₁₂ = 12/2 [2(20)+(n -1)(-1)]

400 = 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 −16)(n −25) = 0

Either (n −16) = 0 or n−25 = 0

n = 16 or n = 25

We know the nth term formula,

a_n = a+(n−1)d

a16 = 20+(16−1)(−1)

a16 = 20−15

a16 = 5

Similarly, the 25th term could be written as;

a25 = 20+(25−1)(−1)

a25 = 20−24

= −4

We can observe that the number of logs in the 16th row is 5 as the numbers cannot be negative.

Hence, 200 logs can be placed in 16 rows and the number of logs in the 16th row is 5.