Find the 6th Term in Sequence: 15, 22.5, 30, ...

Q: How do I know if a sequence is arithmetic?

Calculate the difference between consecutive terms. If all differences are the same, it's arithmetic! For example: 22.5 - 15 = 7.5 and 30 - 22.5 = 7.5, so this sequence is arithmetic.

Q: What if I need to find a term that's not given in the options?

Use the arithmetic sequence formula: $ a_n = a_1 + (n-1) \times d $. Just substitute your values: first term, position number, and common difference.

Q: Can arithmetic sequences have decimal numbers?

Absolutely! Arithmetic sequences can have any type of numbers - whole numbers, decimals, fractions, or even negative numbers. The key is that the difference stays constant.

Q: What's the difference between arithmetic and geometric sequences?

Arithmetic sequences add the same number each time (constant difference), while geometric sequences multiply by the same number each time (constant ratio).

Q: How do I find terms that come before the first given term?

Use the same formula, but with negative values for (n-1). For example, to find the term before 15 in this sequence: $ a_0 = 15 + (0-1) \times 7.5 = 7.5 $.

Arithmetic Sequences with Decimal Terms

Look at the sequence below:

$15,22.5,30,\text{?,?,?}$

What is the 6th element of the sequence?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the 6th term in the sequence

00:05 Notice the change between consecutive terms

00:15 Deduce the next terms from the pattern

00:22 Calculate and substitute

00:44 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 sequence below:

$15,22.5,30,\text{?,?,?}$

What is the 6th element of the sequence?

Step-by-step solution

To determine the 6th element in the sequence, we need to first analyze the pattern of the sequence:

Step 1: Check if the sequence is arithmetic.
Calculate the difference between consecutive terms:

Difference between 22.5 and 15: $22.5 - 15 = 7.5$
Difference between 30 and 22.5: $30 - 22.5 = 7.5$

Both differences are equal to $7.5$ , indicating that the sequence is arithmetic with a common difference of $d = 7.5$ .

Step 2: Find the 6th term of the sequence using the arithmetic sequence formula:
The nth term of an arithmetic sequence is given by $a_n = a_1 + (n-1) \times d$ , where $a_1$ is the first term and $d$ is the common difference.

Calculate the 6th term:
$a_6 = 15 + (6-1) \times 7.5$
$a_6 = 15 + 5 \times 7.5$
$a_6 = 15 + 37.5$
$a_6 = 52.5$

Therefore, the 6th element in the sequence is $52.5$ . This matches option 3 in the provided choices.

Final Answer

$52.5$

Key Points to Remember

Essential concepts to master this topic

Pattern: Arithmetic sequences have a constant difference between consecutive terms
Formula: Use $a_n = a_1 + (n-1) \times d$ where $d = 7.5$
Verify: Check that differences are equal: 22.5-15 = 30-22.5 = 7.5 ✓

Common Mistakes

Avoid these frequent errors

Assuming the sequence follows a different pattern without checking differences
Don't jump to geometric or other patterns without calculating differences first = wrong formula used! This leads to completely incorrect terms. Always find the common difference by subtracting consecutive terms to confirm it's arithmetic.

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 if a sequence is arithmetic?

Calculate the difference between consecutive terms. If all differences are the same, it's arithmetic! For example: 22.5 - 15 = 7.5 and 30 - 22.5 = 7.5, so this sequence is arithmetic.

What if I need to find a term that's not given in the options?

Use the arithmetic sequence formula: $a_n = a_1 + (n-1) \times d$ . Just substitute your values: first term, position number, and common difference.

Can arithmetic sequences have decimal numbers?

Absolutely! Arithmetic sequences can have any type of numbers - whole numbers, decimals, fractions, or even negative numbers. The key is that the difference stays constant.

What's the difference between arithmetic and geometric sequences?

Arithmetic sequences add the same number each time (constant difference), while geometric sequences multiply by the same number each time (constant ratio).

How do I find terms that come before the first given term?

Use the same formula, but with negative values for (n-1). For example, to find the term before 15 in this sequence: $a_0 = 15 + (0-1) \times 7.5 = 7.5$ .

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