Finding the 49-Square Structure in a Geometric Sequence

Q: How do I know if a number can form a square structure?

Check if the number is a perfect square! Take its square root - if you get a whole number, then it can form a square. For example: $ \sqrt{49} = 7 $, so 49 squares make a 7×7 structure.

Q: What does 'element 7' actually mean in this sequence?

Element 7 means the 7th structure in the series, which is a 7×7 square containing $ 7^2 = 49 $ unit squares. The element number always equals the side length!

Q: Could 50 squares form a structure in this series?

No! Since $ \sqrt{50} \approx 7.07 $ is not a whole number, 50 is not a perfect square. Only perfect square numbers can form complete square structures.

Q: How do I find which element has a specific number of squares?

Take the square root of the number of squares. If it's a whole number, that's your element position! For 49 squares: $ \sqrt{49} = 7 $, so it's the 7th element.

Perfect Squares with Geometric Sequence Elements

Here is a series of structures made of squares whose side length is 1 cm.

Is it possible to have a structure in the series that has 49 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 49 squares? If so, at what position?

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

00:29 We see that the number of squares equals the term's position in the sequence

00:38 Therefore we can conclude this is the sequence formula

00:47 We want to find if there is a term with 49 squares

00:52 We'll substitute in the formula and solve for N

00:55 When taking a square root there are always 2 solutions (positive and negative)

00:59 There is no negative position in a sequence, so the positive solution is correct

01:02 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

Here is a series of structures made of squares whose side length is 1 cm.

Is it possible to have a structure in the series that has 49 squares? If so, what element of the series is it?

Step-by-step solution

To determine if it is possible to have a structure in the series with 49 squares, consider the sequence of numbers representing perfect squares: 1, 4, 9, 16, 25, 36, 49, etc. Here, each number represents the total number of 1 cm x 1 cm squares contained in an $n \times n$ square.

The number 49 is indeed a perfect square, since $49 = 7^2$ . This means that a structure made up of 49 squares can indeed be represented as a $7 \times 7$ square.

Therefore, it is possible to have a structure in the series that has 49 squares, and it corresponds to the 7th square in the sequence (since $n = 7$ ), and this matches answer choice Yes, $7$ .

Final Answer

Yes, $7$

Key Points to Remember

Essential concepts to master this topic

Perfect Squares: Each structure contains $n^2$ unit squares forming pattern
Recognition: Check if 49 = $7^2$ , so 7th element works
Verification: Count squares in 7×7 grid: 49 squares total ✓

Common Mistakes

Avoid these frequent errors

Confusing element position with square count
Don't think the 49th element has 49 squares = wrong sequence understanding! The element number equals the side length, not the total squares. Always remember: the nth element is an n×n square with $n^2$ total 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 if a number can form a square structure?

Check if the number is a perfect square! Take its square root - if you get a whole number, then it can form a square. For example: $\sqrt{49} = 7$ , so 49 squares make a 7×7 structure.

What does 'element 7' actually mean in this sequence?

Element 7 means the 7th structure in the series, which is a 7×7 square containing $7^2 = 49$ unit squares. The element number always equals the side length!

Could 50 squares form a structure in this series?

No! Since $\sqrt{50} \approx 7.07$ is not a whole number, 50 is not a perfect square. Only perfect square numbers can form complete square structures.

How do I find which element has a specific number of squares?

Take the square root of the number of squares. If it's a whole number, that's your element position! For 49 squares: $\sqrt{49} = 7$ , so it's the 7th element.

Why does the pattern use perfect squares?

Because we're building square structures! An n×n square always contains exactly $n^2$ unit squares. This creates the sequence: 1, 4, 9, 16, 25, 36, 49...

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