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

Answer:

16 rows and the top row has 5

Step-by-step explanation:

First row = a1 = 20 logs

Common difference = a2 - a1 = 19 - 20 = -1

Find the nth term:

an = a1 + (n - 1)d

an = 20 + (n - 1)(-1)

an = 20 - n + 1

an = 21 - n

Find nth:

Sn = 200

Sn = n/2 (a1 + an)

200 = n/2 (20 + 21 - n)

400 = n(41 - n)

400 = 41n - n²

n² - 41n + 400 = 0

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

n = 16 or n = 25

When n = 16

a16 = 21 - 16

a16 = 5

when n = 25

an = 21 - 25 = -4 (rejected, an cannot be negative)

Answer: There are 16 rows and the top row has 5

Answer 2

Answer:

Total Number of rows required to stack 200 logs = 16

Total number of logs in the 16th row = 5

Step-by-step explanation:

Given that the logs are stacked in rows in the following manner:

No of logs in the bottom row = a₁ = 20

Number of logs in the second row from bottom = a₂ = 19

The number of logs decrease by 1 as we go up from bottom to top, so, if we represent this relation in the form of series we get the following series:

20, 19, 18, 17..............

Now the above series is arithmetic series who common difference (d) can be found as:

d = a₂ - a₁ = 19 - 20 = -1

We need to find

In how many rows 200 logs are stacked = ?

How many logs are in top row = ?

Lets find find the nth term of the above series as show below:

$a_n$ = a₁ + (n - 1)d

$a_n$ = 20 + (n - 1)(-1)

$a_n$ = 20 - n + 1

$a_n$ = 21 - n .......(i)

Now we have to find the total number of rows when total logs become 200.

So, we can take

$S_n$ = 200

Using the summation formula given below:

$S_n$ = $\frac{n}{2}$(a₁ + $a_n$) ...... (A)

Substituting the values of a₁ and $a_n$ into equation (A)

200 = $\frac{n}{2}$ (20 + 21 - n)

2x200 = n(41 - n)

400 = 41n - n²

Rearranging

n² - 41n + 400 = 0

Solving this quadratic equation by middle term breaking

n² - 16n - 25n + 400 = 0

taking commons

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

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

n - 16 = 0 or n - 25 = 0

n = 16 or n = 25

Now, if we take total rows needed to be 25 to stack 200 logs than this negates our sequence as in each row a log must decrease from bottom to top and it is is possible only to have 20 rows only in the series. So 25 rows can not be our correct answer; therefore, our correct answer is 16 rows

So,

Total Number of rows required to stack 200 logs = 16

Now, if we plug the value of n = 16 into equation (i) we get the following expression:

$a_n$ = 21 - 16

$a_n$ = 5

So,

Total number of logs in the 16th row = 5