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
Select the option that represents the correct order of the given words as they would appear in an English dictionary.
1. Decline
2. Decouple
3. Decrease
4. Decorate
5. Decorum
6. Decode
Select the option that represents the correct order of the given words as they would appear in an English dictionary.
1. Penguins
2. Pendants
3. Penthouse
4. Pendulum
5. Pentagon
6. Penalize
Select the option that represents the correct order of the given words as they would appear in an English dictionary.
1. Dissolve
2. Display
3. Distance
4. Disturb
5. Displace
6. Dispute
Which of the following options represents the correct order of the given words as they would appear in the English dictionary?
1. Glob
2. Globule
3. Global
4. Globe
5. Gloom
If the letters of the word ADISNHPANRKARFACHTARKYAR were written in reverse order, then which would be the third letter to the right of the ninth letter from the left?
