Find the Missing Term in Powers Sequence: 729, _, 81, 27, 9, 3

Q: How do I know this is a geometric sequence and not arithmetic?

Check if dividing gives the same result each time! In arithmetic sequences, you subtract to get a common difference. Here, $ \frac{27}{81} = \frac{9}{27} = \frac{1}{3} $, so it's geometric.

Q: Why is the common ratio less than 1?

When the common ratio is between 0 and 1, the sequence decreases! Each term gets smaller because we're multiplying by a fraction like $ \frac{1}{3} $.

Q: What if I can't see the pattern right away?

Start with the known consecutive terms! Look at 81, 27, 9, 3 first. Once you find $ r = \frac{1}{3} $, work backwards to find the missing term.

Q: Can I work backwards from 729 to check my answer?

Absolutely! If 729 × $ \frac{1}{3} $ = 243, and 243 × $ \frac{1}{3} $ = 81, then you know 243 is correct!

Q: What if the sequence had different numbers?

The same method works! Always find the common ratio by dividing consecutive known terms, then use that ratio to find missing terms. The process stays the same regardless of the specific numbers.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing term

00:03 Find the pattern of the sequence

00:11 Calculate the missing term according to the pattern we found

00:20 Break down the multiplication into whole number multiplication and remainder

00:26 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

We wrote numbers on the students' T-shirts according to the constant properties.

729 , _ , 81 , 27 , 9 , 3

What are the numbers written on the T-shirts of the fifth student?

2

Step-by-step solution

To solve this problem, we'll analyze the sequence:

The sequence is given as $729, \_, 81, 27, 9, 3$ .

Step 1: Calculate the common ratio between consecutive terms.
Step 2: Calculate the missing term using the calculated common ratio.

We notice that the known terms in the sequence become smaller, suggesting this is a decreasing series.

Step 1: Identify the common ratio

Check the ratio between consecutive known terms near the missing term:

Calculate the ratio between $81$ and $27$ :

r = \frac{27}{81} = \frac{1}{3}

Calculate the ratio between $27$ and $9$ :

r = \frac{9}{27} = \frac{1}{3}

Calculate the ratio between $9$ and $3$ :

r = \frac{3}{9} = \frac{1}{3}

Each time, the ratio is $\frac{1}{3}$ , confirming that it is indeed constant.

Step 2: Calculate the missing term

Now that we know the common ratio $r = \frac{1}{3}$ , we calculate the missing term between $729$ and $81$ .

\frac{\_}{729} = \frac{1}{3}

Multiplying both sides by $729$ , the missing term is:

\_ = 729 \times \frac{1}{3} = 243

Therefore, the number written on the T-shirt of the fifth student is $243$ .

3

Final Answer

243

Key Points to Remember

Essential concepts to master this topic

Common Ratio: Divide consecutive terms to find the pattern
Technique: Calculate $r = \frac{27}{81} = \frac{1}{3}$ to verify consistency
Check: Verify 729 × $\frac{1}{3}$ = 243 and 243 × $\frac{1}{3}$ = 81 ✓

Common Mistakes

Avoid these frequent errors

Adding or subtracting to find the pattern
Don't look for addition patterns like 729 - 648 = 81! This creates random differences that don't work for the whole sequence. Always divide consecutive terms to find the common ratio in geometric sequences.

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 is a geometric sequence and not arithmetic?

+

Check if dividing gives the same result each time! In arithmetic sequences, you subtract to get a common difference. Here, $\frac{27}{81} = \frac{9}{27} = \frac{1}{3}$ , so it's geometric.

Why is the common ratio less than 1?

+

When the common ratio is between 0 and 1, the sequence decreases! Each term gets smaller because we're multiplying by a fraction like $\frac{1}{3}$ .

What if I can't see the pattern right away?

+

Start with the known consecutive terms! Look at 81, 27, 9, 3 first. Once you find $r = \frac{1}{3}$ , work backwards to find the missing term.

Can I work backwards from 729 to check my answer?

+

Absolutely! If 729 × $\frac{1}{3}$ = 243, and 243 × $\frac{1}{3}$ = 81, then you know 243 is correct!

What if the sequence had different numbers?

+

The same method works! Always find the common ratio by dividing consecutive known terms, then use that ratio to find missing terms. The process stays the same regardless of the specific numbers.

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Find the Missing Term in Powers Sequence: 729, _, 81, 27, 9, 3

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