Question

There are 8436 steel balls, each with a radius of 1 centimeter, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. Determine the number of horizontal layers in the pile ?

• A. 32
• B. 36
• C. 96
• D. 34
• E. 24

Hint:

The sequence of sums of triangular numbers is called Tetrahedral Numbers. Knowing the $n(n+1)(n+2)/6$ formula saves you from manually integrating series progressions during timed exams.

Answer 1

The Correct Option is B

Step 1: Identify the sequence pattern.
The layers form triangular numbers: 1, 3, 6, 10...
The formula for the $n$-th triangular number is $\frac{n(n+1)}{2}$.

Step 2: Use the formula for the sum of triangular numbers.
The total number of balls in a tetrahedral stack of $n$ layers is given by the formula $\frac{n(n+1)(n+2)}{6}$.
We are given the total sum = 8436.
$\frac{n(n+1)(n+2)}{6} = 8436 \Rightarrow n(n+1)(n+2) = 50616$.

Step 3: Solve for $n$ using approximation.
Since $n$, $n+1$, and $n+2$ are very close, $n^{3} \approx 50616$.
We know $30^{3} = 27000$ and $40^{3} = 64000$. The number $n$ is between 30 and 40.
Checking the options, let's test $n = 36$:
$36 \times 37 \times 38 = 50616$. The layers are exactly 36.