Identical balls are tightly arranged in the shape of an equilateral triangle with each side containing n balls. How many balls are there in the arrangement?
n(n + 1) / 2
Balls Arrangement in Equilateral Triangle
The problem asks us to find the total number of identical balls arranged tightly in the shape of an equilateral triangle, where each side contains 'n' balls.
Let's visualize the arrangement for small values of 'n':
- If n = 1, there is 1 ball. The arrangement is just a single point.
- If n = 2, there are 2 balls in the bottom row and 1 ball in the row above it. The total number of balls is 1 + 2 = 3.
- If n = 3, there are 3 balls in the bottom row, 2 balls in the row above that, and 1 ball in the top row. The total number of balls is 1 + 2 + 3 = 6.
- If n = 4, there are 4 balls in the bottom row, 3 in the next, 2 in the next, and 1 in the top row. The total number of balls is 1 + 2 + 3 + 4 = 10.
We can see a pattern here. The number of balls in each row, starting from the top row, is 1, 2, 3, ..., up to n balls in the bottom row.
The total number of balls in the arrangement is the sum of the number of balls in each row.
Total balls = 1 + 2 + 3 + ... + n
This is the sum of the first 'n' natural numbers. The formula for the sum of the first 'k' natural numbers is given by $\frac{k(k+1)}{2}$.
In this case, the total number of balls is the sum of the first 'n' natural numbers, so we use the formula with 'k' replaced by 'n'.
Total number of balls = $\frac{n(n+1)}{2}$
Comparing this with the given options, we find that this formula matches option 2.
Important Questions from Arrangement and Pattern
How many such pairs of digits are there in the number ‘95126139', which have as many digits between them in the number (both forward and backward direction) as they have between them in the Numeric Series?
If all the vowels are dropped from the given arrangement, which of the following is 9th from the right end?
How many such consonants are there which are immediately followed by a consonant and immediately preceded by a vowel?
Which letter is 5th to the right of the 3rd from the left end in the given arrangement?
If in each number, all the three digits are arranged in ascending order within the number, which of the following will be the second lowest number after rearrangement?
