Find the 8th Term in the Arithmetic Sequence: 2, 5, 8, ...

Q: Why do we use (n-1) instead of just n in the formula?

Because the first term already exists! To get from position 1 to position 8, you only need to add the common difference 7 times, not 8 times. Think of it like climbing stairs: to reach the 8th step, you climb 7 steps up.

Q: What if the sequence starts with a negative number?

The formula works exactly the same way! Whether $ a_1 $ is positive, negative, or zero, just substitute it into $ a_n = a_1 + (n-1) \cdot d $.

Q: How can I quickly check if my sequence is really arithmetic?

Calculate the differences between consecutive terms. If all differences are the same number, it's arithmetic! For example: $ 5-2=3 $, $ 8-5=3 $, so the common difference is 3.

Q: Can I just keep adding 3 to find the 8th term?

Yes, but it's time-consuming for large position numbers! Listing out 2, 5, 8, 11, 14, 17, 20, 23 works for the 8th term, but imagine finding the 50th term. The formula is much faster!

Q: What if I need to find which position a certain number is in?

Rearrange the formula! If you know $ a_n = 23 $, solve $ 23 = 2 + (n-1) \cdot 3 $ for n. You'll get $ n = 8 $, confirming 23 is the 8th term.

Arithmetic Sequences with Position Finding

Assuming the sequence continues according to the same rule, what number appears in the 8th element?

$2,5,8\ldots$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the 8th term

00:03 This is the sequence formula

00:06 Let's substitute the appropriate term position in the formula and solve

00:12 Always solve multiplication and division before addition and subtraction

00:18 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

Assuming the sequence continues according to the same rule, what number appears in the 8th element?

$2,5,8\ldots$

Step-by-step solution

To solve for the 8th element in the sequence, follow these steps:

Step 1: Identify the sequence pattern.
Calculate the differences: $5 - 2 = 3$ and $8 - 5 = 3$ . This pattern shows the sequence increases by 3 each time, indicating an arithmetic sequence with common difference $d = 3$ .
Step 2: Use the arithmetic sequence formula $a_n = a_1 + (n-1) \cdot d$ .
Given $a_1 = 2$ , $d = 3$ , and $n = 8$ :
Substitute these values into the formula:
$a_8 = 2 + (8-1) \cdot 3$
$a_8 = 2 + 7 \cdot 3$
$a_8 = 2 + 21$
$a_8 = 23$

Therefore, the 8th element in the sequence is $23$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Use $a_n = a_1 + (n-1) \cdot d$ for arithmetic sequences
Pattern Recognition: Calculate differences: $5 - 2 = 3$ , $8 - 5 = 3$ , so $d = 3$
Verification: Count terms manually: 2, 5, 8, 11, 14, 17, 20, 23 gives 8th term ✓

Common Mistakes

Avoid these frequent errors

Adding the common difference incorrectly to position number
Don't calculate $a_8 = 2 + 8 \cdot 3 = 26$ ! This ignores that the first term already exists, so you only add the difference 7 times, not 8. Always use $(n-1) \cdot d$ in the formula.

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

Why do we use (n-1) instead of just n in the formula?

Because the first term already exists! To get from position 1 to position 8, you only need to add the common difference 7 times, not 8 times. Think of it like climbing stairs: to reach the 8th step, you climb 7 steps up.

What if the sequence starts with a negative number?

The formula works exactly the same way! Whether $a_1$ is positive, negative, or zero, just substitute it into $a_n = a_1 + (n-1) \cdot d$ .

How can I quickly check if my sequence is really arithmetic?

Calculate the differences between consecutive terms. If all differences are the same number, it's arithmetic! For example: $5-2=3$ , $8-5=3$ , so the common difference is 3.

Can I just keep adding 3 to find the 8th term?

Yes, but it's time-consuming for large position numbers! Listing out 2, 5, 8, 11, 14, 17, 20, 23 works for the 8th term, but imagine finding the 50th term. The formula is much faster!

What if I need to find which position a certain number is in?

Rearrange the formula! If you know $a_n = 23$ , solve $23 = 2 + (n-1) \cdot 3$ for n. You'll get $n = 8$ , confirming 23 is the 8th term.

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