Find the Missing First Term in Sequence: ?, 2, -1, -4, -7, -10

Q: How do I know if I found the right common difference?

Check that the same difference works between all consecutive terms! In this sequence: $ -1 - 2 = -3 $, $ -4 - (-1) = -3 $, and so on.

Arithmetic Sequences with Missing First Term

$?,2,-1,-4,-7,-10$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the next term

00:03 Let's examine all terms on the number line

00:10 Let's observe the change between consecutive terms

00:15 We can see the sequence is increasing, and the change is adding 3

00:37 Therefore, using this pattern we'll find the next term

00:46 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

$?,2,-1,-4,-7,-10$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Determine the common difference $d$ of the sequence.
The sequence given is $2, -1, -4, -7, -10$ . Calculate the difference between consecutive terms:

$d = -1 - 2 = -3$ (confirming $-4 - (-1) = -3, -7 - (-4) = -3, -10 - (-7) = -3$ ).

Step 2: Use the common difference to find the term before the first known term (2).

We subtract the common difference from the first known term to find the missing term before it:
The first known term is 2. Calculate $2 - (-3)$ :

$? = 2 + 3 = 5$ .

Therefore, the first term of the sequence is $\boxed{5}$ .

Reviewing the multiple-choice options, the correct choice is $\boxed{5}$ , which corresponds to choice (3): $5$ .

Final Answer

$5$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Find common difference by subtracting consecutive terms
Technique: Subtract common difference from known term: $2 - (-3) = 5$
Check: Verify sequence continues: 5, 2, -1, -4, -7, -10 with d = -3 ✓

Common Mistakes

Avoid these frequent errors

Adding instead of subtracting the common difference
Don't add the common difference to find the previous term = wrong direction! If d = -3 and you do 2 + (-3) = -1, you get the next term instead of the previous one. Always subtract the common difference to go backward in the sequence.

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 I found the right common difference?

Check that the same difference works between all consecutive terms! In this sequence: $-1 - 2 = -3$ , $-4 - (-1) = -3$ , and so on.

Why do I subtract the common difference instead of adding it?

Think of it like walking backward! If each step forward is -3, then to go backward one step, you need to undo that -3 by subtracting it.

What if the common difference is positive?

The method stays the same! If d = +2 and you know the second term is 7, then the first term is $7 - 2 = 5$ . Always subtract d to find the previous term.

How can I double-check my answer?

Write out the complete sequence starting with your answer. If it matches the given terms and keeps the same pattern, you're correct! $5, 2, -1, -4, -7, -10$

What if I get a fraction or decimal as the common difference?

No problem! The same method works. Just be extra careful with your arithmetic when subtracting fractions or decimals.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Series questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime