Ifti is building triangular structures with matchsticks. The first structure needs 3 matchsticks, the second needs 9, and the third needs 18. We need to find a pattern and calculate the number of matchsticks needed for the 20th structure.

1. Problem Description

Ifti is building triangular structures with matchsticks. The first structure needs 3 matchsticks, the second needs 9, and the third needs

1

8. We need to find a pattern and calculate the number of matchsticks needed for the 20th structure.

2. Solution Steps

First, let's analyze the number of matchsticks for the first three structures.

Structure 1: 3 matchsticks

Structure 2: 9 matchsticks

Structure 3: 18 matchsticks

Let's look at the differences between consecutive terms:

9 - 3 = 6

18 - 9 = 9

The differences are not constant, so it's not a linear sequence. Let's look at the ratio of consecutive terms:

9 / 3 = 3

18 / 9 = 2

The ratios are not constant either, so it's not a geometric sequence.

Let's examine the given structures and derive a general formula for the nth structure.

In structure 1, there is 1 large triangle and no smaller triangles. Total sticks = 3 * 1 =

3. In structure 2, there is 1 large triangle divided into 4 smaller triangles. There is 1 large triangle (outer edges), and 3 internal matchsticks. So there are 4 triangles total. The total number of matchsticks is

9. This can be calculated as 3 + 6 =

9. In structure 3, there is 1 large triangle divided into 9 smaller triangles. The total number of matchsticks is

1

8. This can be calculated as 3 + 6 + 9 =

1

8.

So, for structure n, the number of matchsticks is the sum of the first n multiples of 3:

$S_n = 3 + 6 + 9 + ... + 3n$

This is an arithmetic series with first term $a = 3$, common difference $d = 3$, and $n$ terms. We can use the formula for the sum of an arithmetic series:

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

$S_n = \frac{n}{2} [2(3) + (n-1)(3)]$

$S_n = \frac{n}{2} [6 + 3n - 3]$

$S_n = \frac{n}{2} [3n + 3]$

$S_n = \frac{3n(n+1)}{2}$

Now we can calculate the number of matchsticks needed for structure 20:

$S_{20} = \frac{3(20)(20+1)}{2}$

$S_{20} = \frac{3(20)(21)}{2}$

$S_{20} = \frac{1260}{2}$

$S_{20} = 630$

3. Final Answer

The number of matchsticks needed for structure 20 is

6

3

0.