Find The Sum Of Geometric Sequence Calculator

Calculate the sum of the first ‘n’ terms of a geometric sequence using this handy sum of geometric sequence calculator. Enter the first term (a), common ratio (r), and number of terms (n) below.

What is a Sum of Geometric Sequence Calculator?

A sum of geometric sequence calculator is a tool used to find the total sum of a finite number of terms in 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 of 2 and a common ratio of 3. Our sum of geometric sequence calculator helps you add up a specific number of terms from such a sequence without manual calculation.

This calculator is useful for students learning about sequences and series, financial analysts dealing with compound interest or annuities over a fixed period, engineers, and anyone needing to sum a geometric progression quickly. Common misconceptions include confusing it with an arithmetic sequence (where terms are added by a constant difference) or thinking it only applies to infinite series.

Sum of Geometric Sequence Formula and Mathematical Explanation

The sum of the first ‘n’ terms of a geometric sequence (S_n) is given by the formula:

S_n = a(1 – rⁿ) / (1 – r)

Where:

• S_n is the sum of the first ‘n’ terms
• a is the first term
• r is the common ratio
• n is the number of terms

This formula is valid when the common ratio ‘r’ is not equal to 1. If r = 1, then each term is equal to ‘a’, and the sum is simply S_n = n * a.

The sum of geometric sequence calculator uses this formula to compute the result. It first checks if r=1, and if so, uses S_n = n*a. Otherwise, it applies the standard formula.

Variables Table

Variable	Meaning	Unit	Typical Range
a	First Term	Unitless (or same as terms)	Any real number
r	Common Ratio	Unitless	Any real number
n	Number of Terms	Integer	Positive integers (≥ 1)
Sn	Sum of first n terms	Unitless (or same as terms)	Depends on a, r, n

Practical Examples (Real-World Use Cases)

Example 1: Savings Growth

Suppose you save $100 in the first month, and each month you save 10% more than the previous month. This is a geometric sequence with a=100 and r=1.10. How much will you have saved after 6 months?

• First Term (a) = 100
• Common Ratio (r) = 1.10
• Number of Terms (n) = 6

Using the sum of geometric sequence calculator (or formula): S₆ = 100(1 – 1.10⁶) / (1 – 1.10) = 100(1 – 1.771561) / (-0.10) = 100(-0.771561) / (-0.10) = 771.561. You would have saved approximately $771.56.

Example 2: Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 70% of its previous height. What is the total distance the ball travels downwards before it theoretically stops (or after many bounces, say 10)?

The distances it travels downwards are 10, 10*0.7, 10*0.7², …

• First Term (a) = 10
• Common Ratio (r) = 0.7
• Number of Terms (n) = 10 (for 10 downward travels)

Using the sum of geometric sequence calculator: S₁₀ = 10(1 – 0.7¹⁰) / (1 – 0.7) = 10(1 – 0.0282475249) / 0.3 = 10(0.9717524751) / 0.3 ≈ 32.39 meters downwards.

For more on series, see our article on finite vs infinite series.

How to Use This Sum of Geometric Sequence Calculator

1. Enter the First Term (a): Input the initial value of your sequence in the “First Term (a)” field.
2. Enter the Common Ratio (r): Input the constant multiplier between terms in the “Common Ratio (r)” field.
3. Enter the Number of Terms (n): Input how many terms of the sequence you want to sum in the “Number of Terms (n)” field. This must be a positive integer.
4. Calculate: Click the “Calculate Sum” button or simply change the input values. The sum of geometric sequence calculator will update the results automatically.
5. View Results: The primary result (Sum S_n) is highlighted. You’ll also see intermediate values like the nth term, and the formula used.
6. Analyze Chart and Table: The chart visually represents the first few terms and their cumulative sum, while the table lists their values.
7. Reset/Copy: Use “Reset” to go back to default values or “Copy Results” to copy the main findings.

The results from the sum of geometric sequence calculator give you the total after ‘n’ terms. Understanding the common ratio’s impact is crucial.

Key Factors That Affect Sum of Geometric Sequence Results

• First Term (a): The starting value directly scales the sum. A larger ‘a’ leads to a proportionally larger sum.
• Common Ratio (r): This is the most critical factor.
  • If |r| > 1, the terms grow, and the sum increases rapidly with ‘n’.
  • If |r| < 1, the terms shrink, and the sum approaches a limit as 'n' increases (see infinite geometric series).
  • If r is positive, all terms have the same sign as ‘a’.
  • If r is negative, terms alternate in sign.
  • If r = 1, the sum is simply n*a.
  • If r = -1, the sum alternates between ‘a’ and 0.
• Number of Terms (n): The more terms you sum, the larger (or smaller, if terms are negative and |r|>1) the magnitude of the sum becomes, especially when |r| > 1.
• Sign of ‘a’ and ‘r’: The signs of ‘a’ and ‘r’ determine the signs of the terms and thus the sum.
• Magnitude of ‘r’ relative to 1: Whether |r| is greater than, less than, or equal to 1 drastically changes the behavior of the sum as ‘n’ increases.
• Integer vs. Non-integer ‘n’: ‘n’ must be a positive integer representing the number of terms. Our sum of geometric sequence calculator enforces this.

For related concepts, explore our nth term calculator.

Frequently Asked Questions (FAQ)

What is a geometric sequence?

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

How do I find the common ratio (r)?

Divide any term by its preceding term. For example, in 3, 6, 12, 24, the common ratio is 6/3 = 2, or 12/6 = 2.

What if the common ratio (r) is 1?

If r=1, the sequence is a, a, a, …, and the sum of the first n terms is simply n * a. Our sum of geometric sequence calculator handles this case.

What if the common ratio (r) is negative?

The terms of the sequence will alternate in sign. The sum formula still applies.

Can the number of terms (n) be zero or negative?

No, the number of terms ‘n’ must be a positive integer (1, 2, 3, …) when calculating the sum of a finite geometric sequence.

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

In a geometric sequence, you multiply by a common ratio to get the next term. In an arithmetic sequence, you add a common difference.

Can I use this calculator for an infinite geometric series?

This sum of geometric sequence calculator is specifically for a finite number of terms (‘n’). For an infinite geometric series, the sum converges to a(1-r) only if |r| < 1. If |r| >= 1 (and r is not 1), the infinite sum diverges (or is n*a if r=1).

How does this relate to compound interest?

The value of an investment with compound interest over discrete periods can be modeled using a geometric sequence, where the initial principal is ‘a’ and (1 + interest rate) is ‘r’. The sum can represent the total value of deposits growing over time if deposits also grow geometrically.