Find Un For The Sequence Calculator

Calculate the Nth term (Un) of an arithmetic or geometric sequence. Select the sequence type, enter the first term, common difference or ratio, and the term number you want to find.

Sequence Terms Table

Term (n)	Value (Un)
Enter values and calculate to see the table.

Table showing the first 10 terms of the sequence.

What is an Nth Term (Un) Sequence Calculator?

An Nth Term (Un) Sequence Calculator is a tool designed to find the value of a specific term (the 'nth' term, denoted as Un) in a given mathematical sequence, usually either an arithmetic or a geometric sequence. You provide the starting term, the rule governing the sequence (common difference or ratio), and the position of the term you want to find (n), and the calculator determines its value.

This calculator is useful for students learning about sequences, mathematicians, engineers, and anyone dealing with patterns that follow arithmetic or geometric progressions. It helps visualize how sequences grow and predict future values. Our Nth Term (Un) Sequence Calculator handles both arithmetic and geometric sequences.

Common misconceptions include thinking all sequences are either arithmetic or geometric, or that 'n' can be negative or zero in this context (it usually represents the position, starting from 1).

Nth Term (Un) Sequence Calculator Formula and Mathematical Explanation

There are two primary types of sequences this calculator handles:

1. Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

The formula to find the nth term (Un) of an arithmetic sequence is:

Un = a + (n – 1)d

Where:

• Un is the nth term.
• a (or U1) is the first term of the sequence.
• n is the term number (the position of the term in the sequence).
• d is the common difference between terms (d = U2 – U1, U3 – U2, etc.).

2. Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

The formula to find the nth term (Un) of a geometric sequence is:

Un = a * r^{(n – 1)}

Where:

• Un is the nth term.
• a (or U1) is the first term of the sequence.
• n is the term number (the position of the term in the sequence).
• r is the common ratio between terms (r = U2 / U1, U3 / U2, etc.).

Variables Table

Variable	Meaning	Unit	Typical Range
Un	The nth term	(depends on 'a')	Any real number
a (or U1)	First term	(depends on context)	Any real number
d	Common difference	(same as 'a')	Any real number
r	Common ratio	Dimensionless	Any non-zero real number
n	Term number/position	Integer	Positive integers (1, 2, 3…)

Practical Examples (Real-World Use Cases)

Example 1: Arithmetic Sequence

Imagine you start saving $10 (a=10) and decide to save $5 more each week (d=5). How much will you save in the 10th week (n=10)?

• Type: Arithmetic
• First Term (a): 10
• Common Difference (d): 5
• Term Number (n): 10

Using the formula Un = a + (n – 1)d:

U10 = 10 + (10 – 1) * 5 = 10 + 9 * 5 = 10 + 45 = 55

So, in the 10th week, you will save $55.

Example 2: Geometric Sequence

A population of bacteria doubles every hour (r=2). If you start with 50 bacteria (a=50), how many will there be after 6 hours (n=7, because n=1 is at 0 hours, n=2 is after 1 hour, so after 6 hours is n=7)?

• Type: Geometric
• First Term (a): 50
• Common Ratio (r): 2
• Term Number (n): 7 (after 6 hours, meaning the 7th term if n=1 is the start)

Using the formula Un = a * r^{(n – 1)}:

U7 = 50 * 2^{(7 – 1)} = 50 * 2⁶ = 50 * 64 = 3200

After 6 hours, there will be 3200 bacteria.

How to Use This Nth Term (Un) Sequence Calculator

1. Select Sequence Type: Choose either "Arithmetic" or "Geometric" from the dropdown menu.
2. Enter First Term (a): Input the initial value of your sequence.
3. Enter Common Difference (d) or Ratio (r): If you selected "Arithmetic," enter the common difference. If "Geometric," enter the common ratio. The correct input field will be visible.
4. Enter Term Number (n): Specify the position of the term you wish to find (e.g., enter 5 for the 5th term).
5. Calculate: The calculator will automatically update the results as you type, or you can click "Calculate Un".
6. Read Results: The main result (Un) will be highlighted, along with the formula used and the input values.
7. View Table and Chart: The table and chart below the calculator will update to show the first 10 terms of the sequence based on your inputs.
8. Reset: Click "Reset" to clear inputs and return to default values.
9. Copy: Click "Copy Results" to copy the main result, formula, and inputs to your clipboard.

This Nth Term (Un) Sequence Calculator helps you quickly find any term in a sequence without manual calculation.

Key Factors That Affect Nth Term (Un) Results

• First Term (a): The starting point of the sequence directly influences all subsequent terms. A larger 'a' generally leads to larger 'Un' values (assuming d or r > 1 or d>0).
• Common Difference (d): For arithmetic sequences, a larger 'd' results in faster linear growth or decay. A positive 'd' means the sequence increases, negative 'd' means it decreases.
• Common Ratio (r): For geometric sequences, the magnitude of 'r' determines the rate of exponential growth or decay. If |r| > 1, the terms grow rapidly; if 0 < |r| < 1, they decrease towards zero. If r is negative, the terms alternate in sign.
• Term Number (n): The further into the sequence you go (larger 'n'), the more the effect of 'd' or 'r' is compounded, leading to values further from 'a'.
• Sequence Type: The fundamental formula changes between arithmetic (linear growth/decay) and geometric (exponential growth/decay), drastically affecting Un, especially for large 'n'.
• Sign of d or r: A negative 'd' causes the arithmetic sequence to decrease. A negative 'r' causes the geometric sequence to alternate signs, while its magnitude still determines growth or decay.

Understanding these factors helps in predicting the behavior of a sequence and the value of its nth term using the Nth Term (Un) Sequence Calculator.

Frequently Asked Questions (FAQ)

Q: What if my sequence is neither arithmetic nor geometric? A: This Nth Term (Un) Sequence Calculator is specifically for arithmetic and geometric sequences. Other sequences (e.g., Fibonacci, quadratic) have different formulas for their nth term and cannot be calculated here.

Q: Can 'n' be 0 or negative? A: In the context of these standard formulas, 'n' usually represents the position starting from 1 (1st term, 2nd term, etc.), so it's a positive integer. Some definitions might start with n=0, but this calculator assumes n starts from 1.

Q: What happens if the common ratio 'r' is 0 in a geometric sequence? A: If 'r' is 0, all terms after the first one will be 0 (a, 0, 0, 0…). Our Nth Term (Un) Sequence Calculator handles this.

Q: Can the first term 'a' be zero? A: Yes, 'a' can be zero. If 'a' is 0 in an arithmetic sequence, Un = (n-1)d. If 'a' is 0 in a geometric sequence, all terms Un will be 0.

Q: How do I find the sum of a sequence? A: This Nth Term (Un) Sequence Calculator finds a specific term. To find the sum of the first 'n' terms (a series), you would need a series sum calculator.

Q: What if I have the terms but don't know 'd' or 'r'? A: If you have consecutive terms, you can find 'd' by subtracting (d = U2 – U1) or 'r' by dividing (r = U2 / U1). Then use our Nth Term (Un) Sequence Calculator.

Q: Can I use this Nth Term (Un) Sequence Calculator for financial calculations? A: Yes, simple interest can be modeled as an arithmetic sequence, and compound interest (over discrete periods) as a geometric sequence, though dedicated financial calculators are often more suitable.

Q: What does it mean if the common ratio 'r' is negative? A: It means the terms of the geometric sequence alternate in sign (e.g., 2, -4, 8, -16…). The Nth Term (Un) Sequence Calculator correctly handles negative ratios.

Related Tools and Internal Resources

• Arithmetic Sequence Calculator: A tool focused solely on arithmetic sequences, with more details.
• Geometric Sequence Calculator: A dedicated calculator for geometric sequences.
• Series Sum Calculator: Calculates the sum of the first n terms of arithmetic or geometric series.
• Math Calculators: A collection of various mathematical tools.
• Algebra Solver: Helps with various algebraic problems.
• Sequence and Series: Learn more about the concepts of sequences and series.