Arithmetic Sequence: Finding Difference Between 24th and 21st Terms

Q: Why is the answer 9 instead of just subtracting the position numbers 24-21=3?

Great question! The difference between position numbers (24-21=3) tells you how many steps apart they are. But the value difference is 3 steps × 3 (common difference) = 9. You're finding the difference between actual sequence values, not positions!

Q: Can I find the difference without calculating both individual terms?

Yes! Since there are 3 steps between the 21st and 24th terms, and each step increases by 3, the difference is simply 3 × 3 = 9. This shortcut works: (position difference) × (common difference).

Q: How do I identify the common difference in any arithmetic sequence?

Subtract any term from the next term: $ d = a_2 - a_1 $ or $ d = a_3 - a_2 $. In this sequence: 5-2=3 and 8-5=3, so d=3. The difference should be the same between any consecutive terms!

Q: What if I need to find terms that are very far apart, like the 100th term?

The formula $ a_n = a_1 + (n-1)d $ works for any position! For the 100th term: $ a_{100} = 2 + (100-1)(3) = 2 + 297 = 299 $. Large numbers don't change the method!

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

Because the first term has zero steps from itself! Think of it as: start at $ a_1 $, then take (n-1) steps of size d. For $ a_1 $: take 0 steps. For $ a_2 $: take 1 step. This pattern gives us (n-1).

Arithmetic Sequences with Term Differences

What is the difference between the value of the 24th element and the value of the 21st element of the sequence below?

$2,5,8\ldots$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the difference between terms 24 and 21

00:06 Let's observe the constant difference between terms

00:17 Let's count how many terms are between 24 and 21

00:41 We can see that the difference between the numbers equals 3 times the difference

00:48 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

What is the difference between the value of the 24th element and the value of the 21st element of the sequence below?

$2,5,8\ldots$

Step-by-step solution

Given the sequence $2, 5, 8, \ldots$ , we will find the difference between the 24th and 21st terms.

First, identify the parameters of the sequence:
The first term $a_1 = 2$ .
The common difference $d$ is $5 - 2 = 3$ .

The formula for the nth term of an arithmetic sequence is given by:

$a_n = a_1 + (n-1)d$

Let's find the 24th term:

$a_{24} = a_1 + (24-1) \cdot d = 2 + 23 \cdot 3$

$a_{24} = 2 + 69 = 71$

Next, find the 21st term:

$a_{21} = a_1 + (21-1) \cdot d = 2 + 20 \cdot 3$

$a_{21} = 2 + 60 = 62$

Now calculate the difference between these terms:

$a_{24} - a_{21} = 71 - 62 = 9$

Therefore, the difference between the 24th and 21st terms of the sequence is 9.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Use $a_n = a_1 + (n-1)d$ for nth term calculation
Technique: Calculate $a_{24} = 2 + 23(3) = 71$ and $a_{21} = 2 + 20(3) = 62$
Check: Verify common difference is 3 throughout: 5-2=3, 8-5=3 ✓

Common Mistakes

Avoid these frequent errors

Finding individual terms incorrectly by miscounting positions
Don't use n instead of (n-1) in the formula = wrong terms! This shifts every calculation and gives answers that are too large. Always use $a_n = a_1 + (n-1)d$ with (n-1) for the position multiplier.

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

Why is the answer 9 instead of just subtracting the position numbers 24-21=3?

Great question! The difference between position numbers (24-21=3) tells you how many steps apart they are. But the value difference is 3 steps × 3 (common difference) = 9. You're finding the difference between actual sequence values, not positions!

Can I find the difference without calculating both individual terms?

Yes! Since there are 3 steps between the 21st and 24th terms, and each step increases by 3, the difference is simply 3 × 3 = 9. This shortcut works: (position difference) × (common difference).

How do I identify the common difference in any arithmetic sequence?

Subtract any term from the next term: $d = a_2 - a_1$ or $d = a_3 - a_2$ . In this sequence: 5-2=3 and 8-5=3, so d=3. The difference should be the same between any consecutive terms!

What if I need to find terms that are very far apart, like the 100th term?

The formula $a_n = a_1 + (n-1)d$ works for any position! For the 100th term: $a_{100} = 2 + (100-1)(3) = 2 + 297 = 299$ . Large numbers don't change the method!

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

Because the first term has zero steps from itself! Think of it as: start at $a_1$ , then take (n-1) steps of size d. For $a_1$ : take 0 steps. For $a_2$ : take 1 step. This pattern gives us (n-1).

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