Nth Term Calculator
Welcome to the nth term calculator. Quickly find the value of any term in an arithmetic or geometric sequence. Select the sequence type, input the starting values, and the term number you want to find.
Calculate the Nth Term
Formula will be shown here.
First 10 Terms of the Sequence
The table below shows the first 10 terms based on the current inputs.
| Term (n) | Value (aₙ or gₙ) |
|---|
Sequence Growth Chart
Visual representation of the first 10 terms for both sequence types based on inputs.
What is an nth term calculator?
An nth term calculator is a tool used to find the value of a specific term in a sequence, given its position (n), the first term, and the common difference (for arithmetic sequences) or common ratio (for geometric sequences). It helps you determine, for instance, what the 10th, 50th, or 1000th number in a sequence will be without having to list out all the preceding terms.
This calculator is useful for students learning about sequences and series in mathematics, teachers preparing examples, and anyone dealing with patterns that follow arithmetic or geometric progressions.
Common misconceptions include thinking the nth term calculator can find the sum of the terms (that's a series calculator) or that it works for all types of sequences (it's primarily for arithmetic and geometric).
nth Term Formulas and Mathematical Explanation
There are two main types of sequences for which we commonly find the nth term:
1. Arithmetic Sequence
In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference (d).
The formula for the nth term (aₙ) of an arithmetic sequence is:
aₙ = a₁ + (n – 1)d
Where:
- aₙ is the nth term
- a₁ is the first term
- n is the term number
- d is the common difference
2. Geometric Sequence
In a geometric sequence, the ratio between consecutive terms is constant. This constant ratio is called the common ratio (r).
The formula for the nth term (gₙ) of a geometric sequence is:
gₙ = a₁ * r^(n-1)
Where:
- gₙ is the nth term
- a₁ is the first term
- n is the term number
- r is the common ratio
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| aₙ or gₙ | The nth term we want to find | Varies (unitless or depends on context) | Any real number |
| a₁ | The first term of the sequence | Varies | Any real number |
| n | The position of the term in the sequence | Unitless | Positive integers (1, 2, 3, …) |
| d | Common difference (for arithmetic) | Varies | Any real number |
| r | Common ratio (for geometric) | Varies | Any real number (often ≠ 0, 1) |
Practical Examples (Real-World Use Cases)
Example 1: Arithmetic Sequence
Imagine a savings plan where you deposit $50 in the first month and increase your deposit by $10 each subsequent month. We want to find out how much you'll deposit in the 12th month.
- First term (a₁): $50
- Common difference (d): $10
- Term number (n): 12
Using the arithmetic formula: a₁₂ = 50 + (12 – 1) * 10 = 50 + 11 * 10 = 50 + 110 = $160. So, in the 12th month, you would deposit $160.
Example 2: Geometric Sequence
Consider a bacterial culture that starts with 100 bacteria, and the population doubles every hour. We want to find the number of bacteria after 5 hours.
- First term (a₁): 100
- Common ratio (r): 2
- Term number (n): 6 (since after 5 hours is the beginning of the 6th interval, or if we consider hour 0 as term 1, after 5 hours is term 6, but it's easier to think of it as n=5 meaning after 5 doublings, which is term 6 if n=1 is 0 hours) Let's rephrase: after 5 hours, we are looking at the 6th term if the first term is at time 0. Or, if n=1 is after 1 hour, then n=5. Let's assume n=1 is initial, so after 5 hours is n=6. Better: n=1 is initial count (100). After 1 hour (n=2), 200… after 5 hours (n=6).
- Term number (n): We want the population *after* 5 hours, meaning 5 doublings have occurred. If the initial population (at time 0) is the first term (n=1), then after 5 hours is the 6th term (n=6). So n=6.
Using the geometric formula: g₆ = 100 * 2^(6-1) = 100 * 2^5 = 100 * 32 = 3200 bacteria after 5 hours.
How to Use This nth term calculator
- Select Sequence Type: Choose 'Arithmetic' or 'Geometric' from the dropdown.
- Enter First Term (a₁): Input the starting value of your sequence.
- Enter Common Difference (d) or Ratio (r): If 'Arithmetic' is selected, enter the common difference. If 'Geometric' is selected, enter the common ratio. The irrelevant input field will be hidden.
- Enter Term Number (n): Input the position of the term you wish to find (e.g., enter 10 to find the 10th term). This must be a positive integer.
- Calculate: The calculator updates automatically, but you can also click 'Calculate'.
- View Results: The primary result shows the value of the nth term. Intermediate values and the formula used are also displayed.
- Check Table and Chart: The table lists the first 10 terms, and the chart visualizes their growth for both sequence types based on your inputs.
- Reset: Click 'Reset' to clear inputs and go back to default values.
- Copy: Click 'Copy Results' to copy the main result and inputs to your clipboard.
This nth term calculator instantly provides the value you are looking for, along with a table and chart for better understanding.
Key Factors That Affect nth Term Results
- First Term (a₁): This is the starting point. A larger first term will generally lead to larger nth terms (assuming positive d or r > 1).
- Common Difference (d): For arithmetic sequences, a larger positive 'd' means the terms grow faster. A negative 'd' means they decrease.
- Common Ratio (r): For geometric sequences, if |r| > 1, the terms grow or decrease rapidly in magnitude. If 0 < |r| < 1, the terms approach zero. If r is negative, the terms alternate in sign.
- Term Number (n): The further out you go in the sequence (larger 'n'), the more pronounced the effect of 'd' or 'r' becomes.
- Type of Sequence: Geometric sequences with |r| > 1 grow much faster than arithmetic sequences for large 'n'.
- Sign of d or r: A negative 'd' leads to decreasing terms. A negative 'r' leads to alternating signs in the terms.
Understanding these factors helps predict the behavior of a sequence and the results from the nth term calculator.
Frequently Asked Questions (FAQ)
- Q: What is the difference between an arithmetic and a geometric sequence?
- A: An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms.
- Q: Can I use the nth term calculator for sequences that are neither arithmetic nor geometric?
- A: No, this calculator is specifically designed for arithmetic and geometric sequences as they have standard formulas for the nth term. For other sequences, you might need a different approach or a sequence calculator that can identify patterns.
- Q: What if the common ratio 'r' is 0 or 1 in a geometric sequence?
- A: If r=0, all terms after the first are 0. If r=1, all terms are the same as the first term. Our nth term calculator handles these cases.
- Q: What if the common difference 'd' is 0?
- A: If d=0, all terms in the arithmetic sequence are the same as the first term.
- Q: Can 'n' be a fraction or negative?
- A: No, 'n' (the term number) must be a positive integer (1, 2, 3, …) as it represents the position in the sequence.
- Q: How does the nth term calculator handle large numbers?
- A: It uses standard JavaScript number precision. For extremely large numbers resulting from high 'n' and 'r', it might switch to scientific notation or hit precision limits.
- Q: Can I find the sum of the first n terms with this calculator?
- A: No, this is an nth term calculator, not a series sum calculator. You would need a series calculator for that.
- Q: Where can I learn more about arithmetic and geometric progressions?
- A: You can find more information on our pages about arithmetic progression and geometric progression.
Related Tools and Internal Resources
- Arithmetic Sequence Calculator: Focuses solely on arithmetic progressions, including sums.
- Geometric Sequence Calculator: Detailed calculator for geometric progressions and their sums.
- Series Sum Calculator: Calculate the sum of arithmetic or geometric series.
- Math Calculators: A collection of various mathematical tools.
- Algebra Solver: Helps solve algebraic equations and expressions.
- Sequence Formulas: A reference guide to common sequence and series formulas.