Find The Sequence With The Nth Term Calculator

Easily find the value of any term in a sequence using the nth term formula with our nth term calculator. Also see the first few terms visualized.

Calculate the Nth Term

What is an Nth Term Calculator?

An nth term calculator is a tool used to find the value of a specific term in a mathematical sequence, given the formula for the nth term (the general term). It can also be used to list the first few terms of the sequence. A sequence is an ordered list of numbers, and the nth term formula is an expression that defines the value of any term in the sequence based on its position 'n'.

This calculator is useful for students learning about sequences and series, mathematicians, engineers, and anyone dealing with patterns that can be expressed by a formula. For example, if you know the formula for a sequence is `2*n + 1`, you can use the nth term calculator to quickly find the 10th term, 20th term, or any other term without manually calculating all preceding terms.

Common misconceptions include thinking that every sequence has a simple nth term formula (some are defined recursively or by other rules) or that the calculator can derive the formula from a few terms (this calculator requires you to input the formula).

Nth Term Formula and Mathematical Explanation

The "nth term" is a formula or rule that allows you to calculate any term in a sequence using its position number 'n'. For example:

• In an arithmetic sequence like 2, 5, 8, 11…, the first term (a) is 2 and the common difference (d) is 3. The nth term formula is `a + (n-1)d`, which is `2 + (n-1)3 = 3n – 1`.
• In a geometric sequence like 3, 6, 12, 24…, the first term (a) is 3 and the common ratio (r) is 2. The nth term formula is `a * r^(n-1)`, which is `3 * 2^(n-1)`.
• Other sequences can have more complex formulas, like `n^2 + 1`, giving 2, 5, 10, 17…

To use the nth term calculator, you input the formula using 'n' as the variable representing the term number. The calculator substitutes the desired term number for 'n' and evaluates the expression.

The general process is:

1. Identify or input the nth term formula `f(n)`.
2. Specify the term number `n` you want to find.
3. Substitute the value of `n` into the formula `f(n)`.
4. Calculate the result.

Variables Used:

Variable	Meaning	Unit	Typical Range
n	Term number (position in the sequence)	None (integer)	1, 2, 3, … (positive integers)
f(n) or an	The value of the nth term	Depends on the sequence	Any real number
a	First term (in arithmetic/geometric)	Depends on the sequence	Any real number
d	Common difference (in arithmetic)	Depends on the sequence	Any real number
r	Common ratio (in geometric)	Depends on the sequence	Any real number (often non-zero)

Practical Examples (Real-World Use Cases)

Let's see how the nth term calculator can be used.

Example 1: Arithmetic Sequence

Suppose you are saving money, starting with $10 and adding $5 each week. The amount you have at the end of week 'n' follows an arithmetic sequence with the first term a=10 and common difference d=5. The nth term formula is `10 + (n-1)*5` or `5n + 5`.

You want to find out how much money you'll have at the end of week 12 (n=12).

• Formula: `5*n + 5`
• Term n: 12

Using the nth term calculator, input `5*n + 5` and n=12. The 12th term is `5*12 + 5 = 60 + 5 = 65`. You'll have $65 at the end of week 12.

Example 2: Quadratic Sequence

Consider a sequence defined by the formula `n^2 – n + 1`. We want to find the 8th term.

• Formula: `n^2 – n + 1`
• Term n: 8

Input `n^2 – n + 1` (or `n*n – n + 1`) and n=8 into the calculator. The 8th term is `8^2 – 8 + 1 = 64 – 8 + 1 = 57`.

How to Use This Nth Term Calculator

1. Enter the Nth Term Formula: In the "Nth Term Formula" field, type the formula for your sequence using 'n' as the term number. Use standard mathematical operators (+, -, *, /) and '^' for exponents (e.g., `n^2` for n squared). Examples: `3*n + 2`, `n^3 – n`, `100 * (0.5)^(n-1)`.
2. Enter the Term Number (n): In the "Term Number (n) to Find" field, enter the positive integer representing the position of the term you want to calculate (e.g., 5 for the 5th term, 20 for the 20th term).
3. Enter Number of Initial Terms: In the "Number of Initial Terms to Show" field, specify how many terms from the start of the sequence you want to see listed and charted (between 1 and 50).
4. Calculate: Click the "Calculate" button.
5. View Results: The calculator will display:
   • The value of the term you requested.
   • The formula used and the term number.
   • A table showing the first few terms of the sequence and their values.
   • A chart visualizing these initial terms.
6. Reset: Click "Reset" to clear the fields and start over with default values.
7. Copy Results: Click "Copy Results" to copy the main findings to your clipboard.

This nth term calculator is a powerful tool for quickly finding terms in sequences defined by explicit formulas.

Key Factors That Affect Nth Term Results

The value of the nth term is directly determined by:

1. The Formula Itself: The structure of the formula (`linear`, `quadratic`, `exponential`, etc.) dictates how the sequence grows or changes. A linear formula (`an+b`) results in an arithmetic sequence, while an exponential one (`ar^(n-1)`) gives a geometric sequence.
2. The Value of 'n': The term number 'n' directly influences the result. Larger 'n' values generally lead to larger (or smaller, depending on the formula) term values if the sequence is increasing or decreasing.
3. Coefficients and Constants: The numbers within the formula (like the '3' and '2' in `3n+2`) determine the starting point and rate of change of the sequence.
4. The Base and Exponent (for geometric/exponential): In formulas like `a*r^(n-1)`, the base 'r' (common ratio) significantly impacts how quickly the terms grow or shrink. If |r| > 1, terms grow rapidly; if |r| < 1, they approach zero.
5. The Degree of 'n' (for polynomial): In formulas like `n^2 + n + 1`, the highest power of 'n' (the degree) determines the long-term behavior (e.g., quadratic growth).
6. Initial Conditions: Although we input the nth term formula directly, it's often derived from initial conditions like the first term and common difference/ratio. These initial values are embedded within the formula.

Frequently Asked Questions (FAQ)

What if my formula involves n-1?

That's fine. Just enter it as is, for example, `5 + (n-1)*3` or `10 * 2^(n-1)`. The nth term calculator will handle it.

Can this calculator find the formula if I give it a few terms?

No, this calculator requires you to input the nth term formula. Finding the formula from terms is a different problem (sequence prediction or fitting).

What does 'n' represent?

'n' represents the position of a term in the sequence. n=1 is the first term, n=2 is the second term, and so on. It must be a positive integer.

What if the formula gives non-integer results?

The calculator will display the decimal result if the formula produces one. Sequences can have non-integer terms.

Can I use fractions in the formula?

Yes, you can use decimal representations of fractions (e.g., 0.5 instead of 1/2) or use division (e.g., `(n+1)/2`).

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers (terms). A series is the sum of the terms of a sequence. This is an nth term calculator for sequences, not a series calculator for sums.

How do I enter powers like n cubed?

Use the caret symbol `^`, so n cubed would be `n^3`.

What if my sequence starts from n=0?

This calculator assumes sequences start from n=1. If your formula is for a sequence starting at n=0, you might need to adjust 'n' in your formula (e.g., replace 'n' with '(n-1)' if you want the first term here to correspond to n=0 there, but it's usually easier to adapt the formula to start at n=1 for this calculator).