Square Pattern Sequence: Finding the 64-Square Structure Position

Q: How can I double-check my answer?

Square your position number! If you think it's the 8th structure, calculate $ 8^2 = 8 \times 8 = 64 $. If this matches the target number, you're correct!

Q: What's the difference between the position and total squares?

The position tells you which structure in the sequence (1st, 2nd, 3rd...). The total squares is position squared. So the 8th structure has $ 8^2 = 64 $ squares.

Perfect Square Patterns with Sequential Structure Identification

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 64 squares? If so, what element of the series is it?

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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 which position?

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

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

00:39 Therefore we can conclude this is the sequence formula

00:45 We want to find if there's a term with 64 squares

00:49 Let's substitute in the formula and solve for N

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

00:55 N must be positive, there's 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 series of structures formed by squares with side lengths of 1 cm.

Is it possible to have a structure in the series that has 64 squares? If so, what element of the series is it?

Step-by-step solution

To determine the sequence that forms these structures, observe the pattern that these are square grids building up: 1-by-1, 2-by-2, 3-by-3. This creates the series $1^2, 2^2, 3^2, \ldots, n^2$ . We are asked to verify if 64 can be represented in this series.

Let's solve $n^2 = 64$ to find out which structure provides 64 squares:

Start by taking the square root of both sides: $\sqrt{n^2} = \sqrt{64}$ .
This simplifies to $n = 8$ , since $\sqrt{64} = 8$ .

This confirms that the structure with 64 squares is indeed possible and corresponds to the 8th element in our series.

Therefore, the element of the series with 64 squares is $8$ .

Final Answer

Yes, $8$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Square grids follow the sequence 1², 2², 3², ..., n²
Technique: To find position, solve n² = 64, so n = √64 = 8
Check: Verify that 8² = 8 × 8 = 64 squares in the structure ✓

Common Mistakes

Avoid these frequent errors

Confusing position with number of squares
Don't think the 64th structure has 64 squares = wrong interpretation! The position number gets squared to find total squares. Always remember: the nth structure has n² squares, not n squares.

Practice Quiz

Test your knowledge with interactive questions

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

FAQ

Everything you need to know about this question

How do I know this is a perfect square sequence?

Look at the pattern in the diagram: 1×1 grid (1 square), 2×2 grid (4 squares), 3×3 grid (9 squares). Each structure is an n×n square grid, giving us $n^2$ total squares.

What if the number isn't a perfect square?

If you can't find a whole number when taking the square root, then no structure exists with that many squares. For example, 50 squares is impossible because $\sqrt{50}$ isn't a whole number.

Why do we take the square root of 64?

We need to find which position n gives us 64 squares. Since each structure has $n^2$ squares, we solve $n^2 = 64$ by taking the square root: $n = \sqrt{64} = 8$ .

How can I double-check my answer?

Square your position number! If you think it's the 8th structure, calculate $8^2 = 8 \times 8 = 64$ . If this matches the target number, you're correct!

What's the difference between the position and total squares?

The position tells you which structure in the sequence (1st, 2nd, 3rd...). The total squares is position squared. So the 8th structure has $8^2 = 64$ squares.

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