It is observed that 1 triangle can be found in the 1st figure, 3 triangles in the 2nd figure, 6 triangles in the 3rd figure, and so on. John claims that more than 100 triangles can be found in the 15th figure? Do you agree? Explain your answer.

well, the nth figure has n(n+1)/2 triangles, so check it out.

To determine if John's claim is true, let's analyze the pattern based on the given information:

In the 1st figure, there is 1 triangle.
In the 2nd figure, there are 3 triangles.
In the 3rd figure, there are 6 triangles.

Looking at these first few figures, we can see that the number of triangles seems to be increasing by 2 each time. So, we can write the number of triangles as a sequence:

1, 3, 6, ...

To find the nth term of this sequence, we notice that the difference between consecutive terms is increasing by 2 each time:

3 - 1 = 2
6 - 3 = 3
...

We can identify this as an arithmetic sequence with a common difference of 2. The formula to find the nth term of an arithmetic sequence is:

nth term = first term + (n - 1) * common difference

So, for the nth figure, we can use this formula to find the number of triangles:

nth term = 1 + (n - 1) * 2
nth term = 1 + 2n - 2
nth term = 2n - 1

Now, let's substitute n = 15 into the formula to find the number of triangles in the 15th figure:

15th term = 2 * 15 - 1
15th term = 29

Therefore, according to the pattern we have observed, there are 29 triangles in the 15th figure. Since 29 is less than 100, John's claim that there are more than 100 triangles in the 15th figure is incorrect.

To determine the number of triangles in each figure, we can analyze the patterns and use a systematic approach:

In the first figure, there is only one triangle.
In the second figure, we can see three triangles: the big triangle formed by the three smaller triangles.
In the third figure, there are six triangles: the big triangle, plus the three smaller triangles (each composed of two triangles).
Now let's continue this pattern and count the number of triangles in the 15th figure:

Looking at the figure directly is not the most efficient way to count, so let's analyze the pattern and derive a formula.
From observation, we can see that the number of triangles in each figure is double the number of triangles in the previous figure, except the first figure, which only has one triangle. This suggests an exponential growth pattern.

Let's denote the number of triangles in the nth figure as Tn.
From the observation, we have:
T2 = 2 x T1
T3 = 2 x T2
T4 = 2 x T3
...
Tn = 2 x T(n-1)

The first figure has T1 = 1 triangle.
Now, let's calculate the number of triangles in the 15th figure using the formula derived:

T15 = 2 x T14
= 2 x (2 x T13)
= 2 x (2 x (2 x T12))
= ...
= 2^14 x T1
= 2^14 x 1

Calculating, we find:
T15 = 2^14 = 16,384

According to our calculation, the 15th figure has 16,384 triangles.

Therefore, John's claim that there are more than 100 triangles in the 15th figure is correct.