Find The Geometric Sequence Calculator

Calculate Geometric Sequence

First few terms of the sequence

Term (n)	Value (a_n)

What is a geometric sequence calculator?

A geometric sequence calculator is a tool used to analyze and determine the properties of a geometric sequence (also known as a geometric progression). 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.

For example, the sequence 2, 6, 18, 54, … is a geometric sequence with a first term (a) of 2 and a common ratio (r) of 3. Our geometric sequence calculator helps you find:

• The value of a specific term (the nth term) in the sequence.
• The sum of the first 'k' terms of the sequence.
• A preview of the sequence's initial terms.

Anyone studying sequences in mathematics, or dealing with scenarios involving exponential growth or decay (like compound interest, population growth, or radioactive decay at discrete intervals) can benefit from using a geometric sequence calculator.

Common misconceptions include confusing it with an arithmetic sequence (where terms are added by a constant difference, not multiplied by a ratio) or thinking the common ratio must always be greater than 1.

Geometric Sequence Formula and Mathematical Explanation

The core formulas used by the geometric sequence calculator are:

1. The nth term (a_n) of a geometric sequence:

   a_n = a * r^(n-1)

   Where:
   • a_n is the nth term.
   • a is the first term.
   • r is the common ratio.
   • n is the term number.
2. The sum of the first k terms (S_k) of a geometric sequence:

   If r ≠ 1: S_k = a(1 – r^k) / (1 – r)

   If r = 1: S_k = a * k

   Where:
   • S_k is the sum of the first k terms.
   • a is the first term.
   • r is the common ratio.
   • k is the number of terms to sum.

Variables Table

Variable	Meaning	Unit	Typical Range
a	First Term	Unitless or context-dependent	Any real number
r	Common Ratio	Unitless	Any real number (r ≠ 0)
n	Term Number for nth term	Unitless	Positive integer (1, 2, 3, …)
k	Number of Terms to Sum	Unitless	Positive integer (1, 2, 3, …)
a_n	nth Term Value	Same as 'a'	Calculated
S_k	Sum of First k Terms	Same as 'a'	Calculated

Practical Examples (Real-World Use Cases)

Using a geometric sequence calculator can be helpful in various scenarios:

Example 1: Investment Growth

Suppose you invest $1000 (a=1000) and it grows by 5% each year. This means the common ratio is 1.05 (r=1.05). You want to know the value of your investment after 10 years (the 11th term, n=11, since the first term is at year 0 or end of year 1 depending on interpretation, let's say we want the value at the end of the 10th year, which is the 11th term if a is year 0 value, or 10th if a is year 1 value. If a is at start, n=11 for end of 10 years). Let's find the value at the start of the 10th year (n=10).

• a = 1000
• r = 1.05
• n = 10

The 10th term a_10 = 1000 * (1.05)^(10-1) = 1000 * (1.05)^9 ≈ $1551.33. The geometric sequence calculator would give you this value.

Example 2: Depreciation

A machine bought for $50,000 (a=50000) depreciates by 15% each year. The remaining value is 85% of the previous year's value, so r=0.85. What is its value after 5 years (n=6, if year 0 is the first term)?

• a = 50000
• r = 0.85
• n = 6

The 6th term a_6 = 50000 * (0.85)^(6-1) = 50000 * (0.85)^5 ≈ $22185.27. Our find geometric sequence tool helps here.

How to Use This Geometric Sequence Calculator

1. Enter the First Term (a): Input the initial value of your sequence.
2. Enter the Common Ratio (r): Input the constant factor by which each term is multiplied.
3. Enter the Term Number to Find (n): Specify which term number you are interested in (e.g., 5 for the 5th term). This must be a positive integer.
4. Enter the Number of Terms to Sum (k): Specify how many terms from the beginning of the sequence you want to sum up. This also must be a positive integer.
5. View Results: The calculator automatically updates and shows the nth term, the sum of the first k terms, a preview of the sequence, and the formulas used. The table and chart also update.
6. Reset: Click "Reset" to return to the default values.
7. Copy Results: Click "Copy Results" to copy the main outputs and inputs to your clipboard.

Understanding the results helps you see how quickly the sequence grows or shrinks based on 'r'. For more about sequences, see our math calculators page.

Key Factors That Affect Geometric Sequence Results

• First Term (a): This is the starting point. A larger 'a' will scale the entire sequence proportionally.
• Common Ratio (r): This is the most critical factor.
  • If |r| > 1, the sequence grows exponentially (diverges).
  • If |r| < 1, the sequence shrinks towards zero (converges to 0).
  • If r = 1, all terms are the same (a, a, a, …).
  • If r = -1, terms alternate between a and -a.
  • If r < -1, terms grow in magnitude but alternate sign.
  • If -1 < r < 0, terms shrink towards zero and alternate sign.
• Number of Terms (n or k): The further you go in the sequence (larger n or k), the more pronounced the effect of 'r' becomes, especially when |r| > 1.
• Sign of 'a' and 'r': The signs determine if the terms are positive, negative, or alternating.
• Magnitude of 'r': The closer |r| is to 1 (but not 1), the slower the growth or decay. The further from 1, the faster it changes. Compare with our exponential growth calculator.
• Integer vs. Non-Integer Values: While 'n' and 'k' must be positive integers for term number and count, 'a' and 'r' can be any real numbers (though 'r' is non-zero).

For financial applications, 'r' often relates to (1 + interest rate) or (1 – depreciation rate). See our compound interest calculator for a related concept.

Frequently Asked Questions (FAQ)

What if the common ratio (r) is 1?

If r=1, the sequence is constant (a, a, a, …). The nth term is 'a', and the sum of k terms is 'a * k'. Our calculator handles this case.

What if the common ratio (r) is 0?

If r=0, after the first term 'a', all subsequent terms are 0 (a, 0, 0, …). The sum for k>1 is just 'a'.

What happens if the common ratio (r) is negative?

The terms of the sequence will alternate in sign (e.g., 2, -4, 8, -16, … if a=2, r=-2).

Can 'n' or 'k' be zero or negative?

In the standard definition of sequences, the term number 'n' and the number of terms 'k' are positive integers (1, 2, 3, …).

How does a geometric sequence relate to exponential growth?

A geometric sequence is a discrete form of exponential growth (if |r|>1) or decay (if |r|<1). The formula a_n = a * r^(n-1) is an exponential function of n-1.

What's the difference between an arithmetic and a geometric sequence?

In an arithmetic sequence, you add a constant difference to get the next term. In a geometric sequence, you multiply by a constant ratio. Check our arithmetic sequence calculator.

How is the geometric sequence calculator used in finance?

It can model compound interest at discrete intervals, the future value of investments growing at a constant rate per period, or the depreciating value of an asset. See our investment calculator.

What about an infinite geometric series?

If |r| < 1, the sum of an infinite geometric series converges to S = a / (1 - r). This calculator deals with the sum of a finite number of terms (k).

Related Tools and Internal Resources

• Arithmetic Sequence Calculator: Calculates terms and sums for sequences with a common difference.
• Compound Interest Calculator: Shows how investments grow with compounding, related to geometric sequences.
• Exponential Growth Calculator: Models continuous exponential growth, a concept close to geometric progression.
• Math Calculators: A collection of various mathematical tools.
• Finance Calculators: Tools for financial planning and analysis.
• Investment Calculator: Project investment growth over time.