Find the Element with 100 Squares in a Growing Geometric Pattern

To determine in which element in the sequence there are 100 squares, we need to identify the pattern of the sequence.

Let's denote $n$ as the position in the sequence and $S(n)$ as the number of squares in the nth element.

Considering the structural pattern:

• The first element (a single square): $S(1) = 1$
• The second element (form a 2x2 square = 4 squares): $S(2) = 4$
• The third element (form a 3x3 square = 9 squares): $S(3) = 9$
• The fourth element (form a 4x4 square = 16 squares): $S(4) = 16$

From this, we observe that: $S(n) = n^2$. This indicates that the number of squares in the nth element is $n^2$.

We want to find $n$ such that $n^2 = 100$.

Solving the equation $n^2 = 100$, we take the square root of both sides:

Therefore, the element in the sequence which contains 100 squares is the 10th element.

Thus, the solution to the problem is $n = 10$.