Square Pattern Analysis: Finding the 81-Square Structure in a Sequence

Q: How can I tell this is a perfect square sequence?

Look at the pattern: 1st structure has 1 square, 2nd has 4 squares, 3rd has 9 squares. These are 1², 2², 3² - all perfect squares!

Q: What if the square root isn't a whole number?

If √n isn't a whole number, then n cannot be in this sequence. For example, √50 ≈ 7.07, so 50 squares is impossible in this pattern.

Q: Do I need to draw all the structures to find the answer?

No! Once you identify the pattern as n², just calculate: if you want 81 squares, find √81 = 9. The 9th structure will have exactly 81 squares.

Q: How do I know which structure number has a certain amount of squares?

Use the formula: if structure n has $ n^2 $ squares, then to find which structure has k squares, calculate $ n = \sqrt{k} $.

Q: What does the SVG diagram show exactly?

The diagram shows the first 3 structures: 1st: 1×1 = 1 square2nd: 2×2 = 4 squares3rd: 3×3 = 9 squares

Square Number Sequences with Perfect Square Recognition

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 81 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 81 squares? If so, at which position?

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

00:23 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

00:40 We want to find if there's a term with 81 squares

00:43 We'll substitute in the formula and solve for N

00:47 We'll take the square root to isolate N

00:50 When taking a square root there are always 2 solutions (positive and negative)

00:53 In a sequence there can't be a negative position, so the positive solution is correct

00:57 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 81 squares? If so, what element of the series is it?

Step-by-step solution

To solve this problem, we take the following steps:

Step 1: Identify the series pattern.
The provided structures correspond to perfect squares: 1, 4, 9, 16, etc.
Step 2: Determine if 81 is in this series.
Since the number of squares follows $n^2$ , we calculate whether 81 is a perfect square:
$n^2 = 81$ implies $n = \sqrt{81}$ .
Step 3: Calculate the square root.
Calculating $\sqrt{81}$ yields $n = 9$ .
Step 4: Verify the solution.
Since 9 is a natural number and $9^2 = 81$ , the ninth structure in the series has 81 squares.

Therefore, it is possible to have a structure in the series with 81 squares, and it is the ninth element of the series, explicitly identified by computing the sequence.

Based on the choices provided, the correct answer is: Yes, $9$ .

Final Answer

Yes, $9$

Key Points to Remember

Essential concepts to master this topic

Pattern: Sequence follows perfect squares: 1², 2², 3², 4²...
Technique: Find square root of target: √81 = 9
Check: Verify by squaring the position: 9² = 81 ✓

Common Mistakes

Avoid these frequent errors

Assuming the sequence adds squares instead of counting total squares
Don't think structure 3 has 1+4 squares = 5 total! This misses that each structure IS a perfect square arrangement. Always recognize that position n contains exactly n² unit 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 is a perfect square sequence?

Look at the pattern: 1st structure has 1 square, 2nd has 4 squares, 3rd has 9 squares. These are 1², 2², 3² - all perfect squares!

What if the square root isn't a whole number?

If √n isn't a whole number, then n cannot be in this sequence. For example, √50 ≈ 7.07, so 50 squares is impossible in this pattern.

Do I need to draw all the structures to find the answer?

No! Once you identify the pattern as n², just calculate: if you want 81 squares, find √81 = 9. The 9th structure will have exactly 81 squares.

How do I know which structure number has a certain amount of squares?

Use the formula: if structure n has $n^2$ squares, then to find which structure has k squares, calculate $n = \sqrt{k}$ .

What does the SVG diagram show exactly?

The diagram shows the first 3 structures:

1st: 1×1 = 1 square
2nd: 2×2 = 4 squares
3rd: 3×3 = 9 squares

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