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
Study the following information carefully to answer the question given below-
B S 5 % T P 2 @ K L 0 Y 8 N G H * 9 X D 3 & J @ C M ! 1 A ^ E V Z < U $ O Q W N
Which of the following is seventeen to the left of nine from the right?- Each of the digits in the number 4512689 is arranged in ascending order from left to right. The position(s) of how many digits will remain unchanged as compared to that in the original number?
- Each of the digits in the number 7812345 is arranged in ascending order from left to right. The position(s) of how many digits will remain unchanged as compared to that in the original number?
Refer to the following number and symbol series and answer the question that follows. Counting to be done from left to right only. (Note: All numbers are single digit numbers only).
(Left) 9 # 8 $ 1 * £ 3 4 7 + Ω 6 2 & % 5 @ (Right)
How many such symbols are there each of which is immediately preceded by another symbol and also immediately followed by another symbol?Refer to the following number and symbol series and answer the question that follows.
Counting to be done from left to right only. All numbers are single-digit numbers.
(Left) 4 6 % Ω 9 # @ 8 $ 7 1 * £ 5 & + 3 2 (Right)
How many such symbols are there each of which is immediately preceded by a number and also immediately followed by another symbol?
