Square Pattern Sequence: Finding the Number of Squares in the 5th Element

Q: How do I know this follows an n² pattern?

Look at the progression: 1st element has 1 square, 2nd has 4 squares, 3rd has 9 squares. These are perfect squares: $ 1^2, 2^2, 3^2 $, so the pattern is $ n^2 $!

Q: What if I can't see all the squares clearly in the diagram?

Focus on the dimensions of each square figure. The 1st is 1×1, 2nd is 2×2, 3rd is 3×3. So the 5th element would be 5×5 = 25 squares.

Q: How do I count squares without making mistakes?

Break it into rows and columns. A 5×5 square has 5 rows, each with 5 unit squares, so $ 5 \times 5 = 25 $ total squares.

Quadratic Pattern Recognition with Perfect Squares

Below is a sequence represented by squares. How many squares will there be in the 5 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 fifth term in the sequence.

00:10 First, let's count the squares in each term. Ready? Let's go!

00:32 Notice how the number of squares is the same as the term's position squared.

00:42 This means our sequence's formula is: position number, squared.

00:49 Now, let's substitute the position of the term and do the math.

00:57 And there you have it. That's the solution!

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 5 element?

Step-by-step solution

To solve the problem, we will analyze the sequence of element growth:

Step 1: Identify the pattern in the sequence from the image.
- Normally, a sequence of squares that increases in size might do so according to $n^2$ (a perfect square sequence).
- Observing the sequence, we see that the first element has $1^2 = 1$ square, the second element has $2^2 = 4$ squares, the third has $3^2 = 9$ squares, and the fourth follows similarly.
Step 2: Apply the identified pattern to compute the 5th element of the sequence.
- When following the pattern $n^2$ , the 5th element would naturally contain $5^2 = 25$ squares.

Thus, the number of squares in the 5th element of the sequence is $25$ .

Final Answer

$25$

Key Points to Remember

Essential concepts to master this topic

Pattern: Identify that each element follows $n^2$ square sequence
Technique: Count unit squares: 1st=1, 2nd=4, 3rd=9, so 5th= $5^2=25$
Check: Verify pattern holds: $1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25$ ✓

Common Mistakes

Avoid these frequent errors

Counting visible grid lines instead of unit squares
Don't count the outer edges or grid lines = wrong total! Grid lines form the boundaries, but you need to count the actual square units inside. Always count the individual unit squares that fill each figure.

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 follows an n² pattern?

Look at the progression: 1st element has 1 square, 2nd has 4 squares, 3rd has 9 squares. These are perfect squares: $1^2, 2^2, 3^2$ , so the pattern is $n^2$ !

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

Focus on the dimensions of each square figure. The 1st is 1×1, 2nd is 2×2, 3rd is 3×3. So the 5th element would be 5×5 = 25 squares.

Could this be a different pattern like adding consecutive numbers?

Always test your pattern! Adding pattern would give: 1, 1+3=4, 4+5=9, 9+7=16. But $n^2$ gives the same results and is simpler: stick with the square pattern.

How do I count squares without making mistakes?

Break it into rows and columns. A 5×5 square has 5 rows, each with 5 unit squares, so $5 \times 5 = 25$ total squares.

What if the 4th element doesn't match n² = 16?

Double-check by carefully counting the 4th element's unit squares. If it's truly 16, then the pattern holds. If not, you might need to reexamine the sequence for a different pattern.

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