Question

200 pepsi bottles are stacked in such a way that there are 20 bottles in the bottom, 19 in the next row, 18 in the row next and so on. in how many rows will 200 pepsi bottles be placed?

Answer 1

16 ROWS ARE REQUIRED TO KEEP THE PEPSI BOTTLE.

Answer 2

The number of rows, in which bottles are placed, is 16.

Given,

200 Pepsi bottles: stacked in some rows, such that:

20 bottles in the bottom row,

19 in the next row, and so on.

To find,

The number of rows to place all 200 bottles.

Solution,

It can be seen that the given situation of stacking 200 Pepsi bottles in different rows, can be modeled as an arithmetic progression or A. P.

Let $n$ be the number of rows in which bottles are stacked.

Now, the A. P. will be,

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

And, for this A. P.

$S_n=200$ (total number of bottles),

$a=20$ (bottles in the bottom or 1st row),

$d=a_2-a_1=19-20=-1$ (difference of the number of bottles in any 2 consecutive rows),

To determine $n$, that is the number of rows, the formula for the sum of A. P. can be used, as follows.

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

Substituting the respective values,

$200=\frac{n}{2} [2(20)+(n-1)(-1)]$

Simplifying,

$400=n[40-n+1]$

$\implies 400=n[41-n]$

$\implies 400=41n-n^2$

$\implies n^2-41n+400=0$

The above quadratic equation can be solved by factorization, as follows.

$n^2-25n-16n+400=0$

$\implies n(n-25)-16(n-25)=0$

$\implies (n-25)(n-16)=0$

⇒ $n-25=0$ or, $n-16=0.$

So,

$n=25$, or $n=16.$

The number of bottles in the $n^{th}$ row can be determined using the formula for the $n^{th}$ term of A. P. that is,

$a_n=a+(n-1)d$

Now, consider $n=25$, then the number of bottles in the 25th row:

$a_{25}=20+(25-1)(-1)=20-24=-4$

$\implies a_{25}=-4$.

Again, consider $n=16$ now, then the number of bottles in the 16th row:

$a_{16}=20+(16-1)(-1)=20-15=5$

$\implies a_{16}=5$.

We can observe from above that, the number of bottles in 25 rows ($n=25$) is coming out to be negative 4, which is not possible.

But, in 16 rows ($n=16$), the number of bottles is 5.

So, the number of rows = 16.

Therefore, the number of rows, in which bottles are placed, is 16.

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