Square Pattern Sequence: Finding the Structure with 64 Units

Q: How do I know this is about n² and not just counting squares?

Look at the pattern! The first structure is 1×1 = 1 square, second is 2×2 = 4 squares, third is 3×3 = 9 squares. Each position n forms an n×n grid.

Q: Why do we take the square root of 64?

Because we need to find which position number gives us 64 squares total. Since position n has n² squares, we solve $ n^2 = 64 $ by taking $ n = \sqrt{64} = 8 $.

Q: How can I double-check my answer without a calculator?

Use your multiplication facts! Since 8 × 8 = 64, you know that 8² = 64. You can also think: 8 rows × 8 columns = 64 small squares.

Q: What would the 10th structure look like?

The 10th structure would be a 10×10 square containing $ 10^2 = 100 $ small unit squares. Each structure gets bigger as a perfect square!

Square Number Sequences with Perfect Square Recognition

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

In which structure (element) of the series are there 64 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 64 squares? If so, at what position?

00:04 Let's count the squares in each term

00:26 We see that the number of squares equals the term's position in the sequence

00:32 Therefore we can conclude this is the sequence formula

00:42 We want to find if there's a term with 64 squares

00:47 We'll substitute in the formula and solve for N

00:50 We'll take the square root to isolate N

00:54 N must be positive, there's no negative position in the sequence

00:58 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 series of structures formed by squares with side lengths of 1 cm.

In which structure (element) of the series are there 64 squares?

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Identify the formula for the sequence.
Step 2: Set up the equation to reflect the total number of squares as n².
Step 3: Solve for n such that n² = 64.

Now, let's work through each step:

Step 1: The sequence in question forms larger squares with each subsequent position based on the SVG graphic provided.

Step 2: We know that the nth position has n² squares: $n^2 = 64$ .

Step 3: Solving for n in the equation $n^2 = 64$ , we take the square root of both sides:

$n = \sqrt{64} = 8$ .

Therefore, the structure with 64 squares occurs at the 8th position in the series.

Thus, the correct answer is $8$ .

Final Answer

$8$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Each structure forms an n×n square with n² total units
Equation Setup: Set n² = 64 and solve for n by taking square root
Verification: Check that 8² = 8×8 = 64 squares matches the target ✓

Common Mistakes

Avoid these frequent errors

Confusing position number with total squares
Don't think position 64 has 64 squares = wrong interpretation! The sequence shows position n has n² squares, not n squares. Always identify that we need n² = 64, so n = 8 is the position number.

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 do I know this is about n² and not just counting squares?

Look at the pattern! The first structure is 1×1 = 1 square, second is 2×2 = 4 squares, third is 3×3 = 9 squares. Each position n forms an n×n grid.

What if 64 isn't a perfect square?

Great question! If the number wasn't a perfect square (like 50), then no structure in this sequence would have exactly that many squares. Always check if $\sqrt{n}$ gives a whole number.

Why do we take the square root of 64?

Because we need to find which position number gives us 64 squares total. Since position n has n² squares, we solve $n^2 = 64$ by taking $n = \sqrt{64} = 8$ .

How can I double-check my answer without a calculator?

Use your multiplication facts! Since 8 × 8 = 64, you know that 8² = 64. You can also think: 8 rows × 8 columns = 64 small squares.

What would the 10th structure look like?

The 10th structure would be a 10×10 square containing $10^2 = 100$ small unit squares. Each structure gets bigger as a perfect square!

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