Find the Element with 100 Squares in a Growing Geometric Pattern

Q: How can I tell this follows an n² pattern just by looking?

Look at the dimensions of each figure! Element 1 is 1×1, element 2 is 2×2, element 3 is 3×3. The number of unit squares equals width × height = n².

Q: Do I always take the positive square root?

Yes! Since we're talking about position numbers in a sequence, negative values don't make sense. Position numbers are always positive integers.

Q: What's the fastest way to solve n² = 100?

Mental math works great here! Think: "What number times itself equals 100?" Since 10 × 10 = 100, the answer is n = 10. For harder cases, use $ n = \sqrt{100} $.

Square Number Patterns with Geometric Sequences

The following is a sequence of structures formed from squares with side lengths of 1 cm.

In which element of the sequence are there 100 squares?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Will there be a term with 100 squares? If so, at which position?

00:05 Let's count the squares in each term

00:25 We can see that the number of squares equals the term's position squared

00:34 Therefore we can conclude this is the sequence formula

00:39 We want to find if there is a term with 100 squares

00:45 Let's substitute in the formula and solve for N

00:52 We'll take the square root to isolate N

00:57 N must be positive, there is no negative position in the sequence

01:00 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The following is a sequence of structures formed from squares with side lengths of 1 cm.

In which element of the sequence are there 100 squares?

Step-by-step solution

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:

n = \sqrt{100} = 10

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

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

Final Answer

$10$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Each element contains n² squares where n is position number
Formula Application: Set n² = 100 and solve: √100 = 10
Verification: Check that 10² = 100 squares matches the target ✓

Common Mistakes

Avoid these frequent errors

Counting individual squares instead of recognizing the n² pattern
Don't try to count every single square in each figure = takes forever and leads to counting errors! Students often miss the geometric relationship. Always look for the underlying pattern: position 1 has 1² squares, position 2 has 2² squares, position n has n² squares.

Practice Quiz

Test your knowledge with interactive questions

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

FAQ

Everything you need to know about this question

How can I tell this follows an n² pattern just by looking?

Look at the dimensions of each figure! Element 1 is 1×1, element 2 is 2×2, element 3 is 3×3. The number of unit squares equals width × height = n².

What if the answer wasn't a perfect square?

If n² = 100 gave us something like n = 7.5, that would mean no element has exactly 100 squares. You'd need to specify which element comes closest or has at least 100 squares.

Do I always take the positive square root?

Yes! Since we're talking about position numbers in a sequence, negative values don't make sense. Position numbers are always positive integers.

How do I know this isn't some other pattern like 2n or n+5?

Check the given examples! Element 1 has 1 square, element 2 has 4 squares, element 3 has 9 squares. These are $1^2, 2^2, 3^2$ , confirming the n² pattern.

What's the fastest way to solve n² = 100?

Mental math works great here! Think: "What number times itself equals 100?" Since 10 × 10 = 100, the answer is n = 10. For harder cases, use $n = \sqrt{100}$ .

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