Square Pattern Sequence: Finding the 55-Square Structure Position

The following is a series of structures formed by squares with side lengths of 1 cm.

Is it possible to have a structure in the series that has 55 squares? If so, what element of the series is it?

To determine whether a structure with 55 squares exists in the sequence, we'll follow these steps:

• Step 1: Determine the pattern or sequence rule from the given sequence of structures.
• Step 2: Develop a formula for the nth term of the sequence.
• Step 3: Check if there's an $n$ such that $a_n = 55$.

Let's work through each step:

Step 1: Analyze the given structures. The observed structure from the problem graphic suggests a pattern in the number of squares. For instance, visualize a small series increasing as follows:
- First structure: $1$ square (1x1)
- Second structure: $1 + 3 = 4$ squares (2x2)
- Third structure: $4 + 5 = 9$ squares (3x3)
- Fourth structure: $9 + 7 = 16$ squares (4x4), and so forth.

Step 2: Recognize the sequence is quadratic in nature (perfect square numbers). The general term structure is the nth square, meaning $a_n = n^2$. Let's verify the sequence progresses by square numbers.

Step 3: Set $n^2 = 55$ and solve for $n$. Solving $n^2 = 55$ to check if $55$ is a perfect square:

$n = \sqrt{55} \approx 7.416$.

Since $n$ should be an integer and 55 isn't a perfect square, $55$ is not a term in this sequence.

Thus, there's no element in the series possessing exactly 55 squares. The final conclusion follows accordingly based on the sequence rule verification.

Therefore, the solution to the problem is No, as there’s no such element that has exactly 55 squares in the series.