Identify Terms in Sequence: Finding Values in an = 3n + 1

Q: Why do I need to solve for n instead of just calculating terms?

Because you're given potential terms and need to check if they belong to the sequence! Solving $ n = \frac{\text{value} - 1}{3} $ tells you exactly which position (if any) would produce that term.

Q: What if I get a decimal or fraction for n?

Then that value is not a term in the sequence! Position numbers must be positive integers (1, 2, 3, 4...). Decimals like 11.67 mean the value falls between two actual terms.

Q: How do I remember the backwards formula?

Start with $ a_n = 3n + 1 $, then solve for n: subtract 1 from both sides, then divide by 3. So $ n = \frac{a_n - 1}{3} $!

Q: What does it mean that 40 is the 13th term?

It means when you substitute $ n = 13 $ into the formula, you get $ a_{13} = 3(13) + 1 = 40 $. The 13th position in the sequence contains the value 40.

Arithmetic Sequences with Integer Position Validation

Given the series, y represents some term in the series and n represents the position of the term in the series.

Only one of the following is a term in the series, reveal it:

$a_n=3n+1$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find which of the following is a member of the sequence

00:04 We'll substitute this solution in the formula and solve for X

00:09 If the solution for X is positive and whole, then it's a member of the sequence

00:12 Let's isolate X

00:19 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

Given the series, y represents some term in the series and n represents the position of the term in the series.

Only one of the following is a term in the series, reveal it:

$a_n=3n+1$

Step-by-step solution

To solve this problem, we need to identify which of the given numbers is a term in the series defined by $a_n = 3n + 1$ .

We will evaluate each choice:

Substitute 36: $n = \frac{36 - 1}{3} = \frac{35}{3}$ . This is not an integer, so 36 is not a term.
Substitute 39: $n = \frac{39 - 1}{3} = \frac{38}{3}$ . This is not an integer, so 39 is not a term.
Substitute 33: $n = \frac{33 - 1}{3} = \frac{32}{3}$ . This is not an integer, so 33 is not a term.
Substitute 40: $n = \frac{40 - 1}{3} = \frac{39}{3} = 13$ . This is an integer, so 40 is a term.

Therefore, the number 40 is a term in the series.

$40$

Final Answer

$40$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $a_n = 3n + 1$ to generate sequence terms
Technique: Solve $n = \frac{\text{value} - 1}{3}$ to check if value exists
Check: Position n must be a positive integer for valid term ✓

Common Mistakes

Avoid these frequent errors

Testing values without solving for n
Don't just plug in random position numbers and calculate terms! This wastes time and misses the point. Always solve backwards: if a term equals some value, then $n = \frac{\text{value} - 1}{3}$ must give a positive integer.

Practice Quiz

Test your knowledge with interactive questions

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

FAQ

Everything you need to know about this question

Why do I need to solve for n instead of just calculating terms?

Because you're given potential terms and need to check if they belong to the sequence! Solving $n = \frac{\text{value} - 1}{3}$ tells you exactly which position (if any) would produce that term.

What if I get a decimal or fraction for n?

Then that value is not a term in the sequence! Position numbers must be positive integers (1, 2, 3, 4...). Decimals like 11.67 mean the value falls between two actual terms.

How do I remember the backwards formula?

Start with $a_n = 3n + 1$ , then solve for n: subtract 1 from both sides, then divide by 3. So $n = \frac{a_n - 1}{3}$ !

Can I just calculate the first few terms to check?

That's very inefficient! For large numbers like 40, you'd need to calculate many terms. The algebraic method gives you the answer instantly: $n = \frac{40-1}{3} = 13$ .

What does it mean that 40 is the 13th term?

It means when you substitute $n = 13$ into the formula, you get $a_{13} = 3(13) + 1 = 40$ . The 13th position in the sequence contains the value 40.

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