Square Sequence Analysis: Finding the 6th Element Pattern

Q: Can I just add the same number each time instead?

No! This isn't an arithmetic sequence. The differences between consecutive terms aren't constant: 4-1=3, 9-4=5, 16-9=7. It follows $n^2$ pattern instead.

Q: How do I calculate 6² quickly in my head?

Break it down: $6^2 = 6 \times 6$. Think 6 × 6 = (5+1) × 6 = 30 + 6 = 36. Or memorize that 6² = 36 since it's a common perfect square!

Perfect Square Patterns with Visual Representation

Below is a sequence represented by squares. How many squares will there be in the 6th element?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find the sixth term in the sequence.

00:10 First, let's count the number of squares in each term.

00:20 Notice, the number of squares equals the term's position, raised to the power of two.

00:32 So, we can conclude that this is the formula for the sequence.

00:39 Now, substitute the term's position to calculate its value.

00:46 And that's how we find the solution to this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Below is a sequence represented by squares. How many squares will there be in the 6th element?

Step-by-step solution

To solve this problem, we'll follow these steps:

Identify the sequence pattern: it follows perfect squares.
Associate each step with its corresponding perfect square number.
Calculate the 6th element's number of squares using the formula $n^2$ .

Now, let's work through each step:
Step 1: The sequence is defined by the perfect square numbers: $1^2, 2^2, 3^2, \ldots$ .
Step 2: Each element in this sequence corresponds to an integer squared, revealing the sequence as $1, 4, 9, 16, 25, \ldots$ .
Step 3: For the 6th element, calculate $6^2$ :
$6^2 = 36$

Therefore, the solution to the problem is 36.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Pattern Rule: Each element equals n² where n is the position number
Technique: Count position carefully: 1st = 1², 2nd = 2², 6th = 6² = 36
Check: Verify by counting actual squares in early elements: 1, 4, 9 matches 1², 2², 3² ✓

Common Mistakes

Avoid these frequent errors

Confusing position number with square count
Don't count squares and use that as the position = wrong formula! Students often count 4 squares and think it's the 4th element when it's actually the 2nd. Always identify the element's position first, then apply n².

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 a perfect square sequence?

Look at the pattern: 1 square, 4 squares, 9 squares... These are 1², 2², 3². The visual shows square grids getting larger in this specific pattern.

What if I can't see the pattern clearly in the diagram?

Count the squares in each element carefully. Write down the numbers: 1, 4, 9... then ask "What operation gives me these results?" You'll see they're perfect squares!

Can I just add the same number each time instead?

No! This isn't an arithmetic sequence. The differences between consecutive terms aren't constant: 4-1=3, 9-4=5, 16-9=7. It follows $n^2$ pattern instead.

How do I calculate 6² quickly in my head?

Break it down: $6^2 = 6 \times 6$ . Think 6 × 6 = (5+1) × 6 = 30 + 6 = 36. Or memorize that 6² = 36 since it's a common perfect square!

What would the 10th element be?

Using the same pattern: $10^2 = 100$ squares. The sequence continues: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

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