To determine if a number is part of a sequence defined by an nth term rule, you need to use a simple algebraic approach. The key is to understand what the nth term actually represents, and use that to solve your problem.
Understanding the nth Term
The nth term is a formula that describes any term in a sequence based on its position. For example, in a sequence like 2, 4, 6, 8..., the nth term might be 2n. This means the 1st term is 2 1 = 2, the 2nd term is 2 2 = 4, and so on.
The Method to Check if a Number is in the Sequence
Here's the step-by-step method to find if a given number exists in a sequence given its nth term:
- Set Up an Equation: Take the nth term formula and set it equal to the number you're testing.
- Solve for n: Solve the resulting equation for 'n'.
- Check for Integer Solution: If 'n' is an integer (a whole number), the number is in the sequence. If 'n' is not an integer, the number is not in the sequence.
| Step | Description | Example (Using 2n and testing for 10) |
|---|---|---|
| 1 | Set the nth term equal to the number being tested. | 2n = 10 |
| 2 | Solve the equation for 'n' | n = 10 / 2 => n = 5 |
| 3 | Is 'n' an integer? | Yes, n=5 is an integer |
| Result | Number is in the sequence if n is an integer | 10 is in the sequence 2, 4, 6... |
Practical Examples
Let's consider an example of where we would use this method:
- Example 1: If a sequence's nth term is 3n + 1, is 16 part of the sequence?
- Equation: 3n + 1 = 16
- Solve for n: 3n = 15, so n = 5
- Result: Yes, 16 is the 5th term in the sequence.
- Example 2: If a sequence's nth term is n² - 1, is 20 part of the sequence?
- Equation: n² - 1 = 20
- Solve for n: n² = 21, so n = √21 which is about 4.58
- Result: No, as the square root of 21 is not an integer, therefore 20 is not a term in the sequence.
Why This Method Works
The reference states: "We can use the n th term to work out whether a number is in a sequence by putting the n th term equal to the number and solving the equation to find n . Because n is the term number it has to be an integer (a whole number)."
- This is because 'n' represents the position or term number in the sequence. Therefore, it can't be a fraction or a decimal. If the number falls between terms, it won't be a part of the sequence.
Key Takeaways
- You can effectively use the nth term formula to test any number's membership in the sequence.
- The core idea is to determine if a whole number solution for 'n' exists.
- This algebraic technique is versatile and applicable to different types of sequences.
