Uncover the Property in the Number Set: 8, 12, 18, 27

Q: How do I know which pattern to look for first?

Start with arithmetic patterns (adding/subtracting the same number) since they're simpler. If differences aren't equal, try geometric patterns (multiplying/dividing by the same number) by calculating ratios.

Q: What if the ratios aren't exact whole numbers like 2 or 3?

That's totally normal! Ratios can be decimals like $ 1.5 $ or fractions like $ \frac{3}{2} $. Just make sure all ratios are exactly the same for a geometric sequence.

Q: Could there be other types of patterns besides arithmetic and geometric?

Yes, but arithmetic and geometric sequences are the most common in basic math. Always check these two first before considering more complex patterns like quadratic or exponential sequences.

Q: What does 'multiply by 1.5' actually mean?

Multiplying by $ 1.5 $ means making each number 1.5 times bigger than the previous one. It's the same as adding half the number to itself: $ 8 + 4 = 12 $.

Number Sequences with Ratio Patterns

Look at the following set of numbers and determine if there is any property, if so, what is it?

$8,12,18,27$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is there any pattern? And if so, what is it?

00:03 Let's observe the change between terms

00:16 We can see that the pattern is constant and it's multiplying by 1.5

00:25 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

Look at the following set of numbers and determine if there is any property, if so, what is it?

$8,12,18,27$

Step-by-step solution

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

Step 1: Determine if there's a consistent pattern between the numbers
Step 2: Test for arithmetic progression
Step 3: Test for geometric progression
Step 4: Choose the correct option based on calculations

Now let's process each step:

Step 1: Check for Arithmetic Progression

The differences between the numbers are calculated as follows:

$12 - 8 = 4$
$18 - 12 = 6$
$27 - 18 = 9$

The differences are not equal ( $4, 6, \text{and } 9$ ), so it's not an arithmetic sequence.

Step 2: Check for Geometric Progression

Calculate the ratios between consecutive numbers:

$\frac{12}{8} = 1.5$
$\frac{18}{12} = 1.5$
$\frac{27}{18} = 1.5$

The ratios are the same ( $1.5$ ), indicating a geometric progression where each number is multiplied by $1.5$ .

Conclusion: The numbers $8, 12, 18, 27$ form a geometric sequence with a common ratio of $\times 1.5$ .

Therefore, the property of the sequence is $\times 1.5$ .

Final Answer

$\times1.5$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Test both arithmetic differences and geometric ratios systematically
Technique: Calculate ratios: 12÷8 = 1.5, 18÷12 = 1.5, 27÷18 = 1.5
Check: Verify each term equals previous term × 1.5: 8×1.5=12 ✓

Common Mistakes

Avoid these frequent errors

Assuming no pattern exists after checking differences only
Don't stop after finding unequal differences like 4, 6, 9 and conclude no pattern exists! This ignores geometric sequences entirely. Always check ratios next: divide each term by the previous one to find if there's a multiplication pattern.

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 which pattern to look for first?

Start with arithmetic patterns (adding/subtracting the same number) since they're simpler. If differences aren't equal, try geometric patterns (multiplying/dividing by the same number) by calculating ratios.

What if the ratios aren't exact whole numbers like 2 or 3?

That's totally normal! Ratios can be decimals like $1.5$ or fractions like $\frac{3}{2}$ . Just make sure all ratios are exactly the same for a geometric sequence.

Could there be other types of patterns besides arithmetic and geometric?

Yes, but arithmetic and geometric sequences are the most common in basic math. Always check these two first before considering more complex patterns like quadratic or exponential sequences.

What does 'multiply by 1.5' actually mean?

Multiplying by $1.5$ means making each number 1.5 times bigger than the previous one. It's the same as adding half the number to itself: $8 + 4 = 12$ .

How can I double-check my geometric sequence answer?

Work backwards! Start with your ratio and see if you can generate the sequence: $8 \times 1.5 = 12$ , $12 \times 1.5 = 18$ , $18 \times 1.5 = 27$ ✓

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