Find The Sum Of The Geometric Sequence Calculator

Calculate the Sum (S_n)

Enter the details of your geometric sequence below.

First 10 terms (or n terms if n < 10) of the sequence:

Term No.	Term Value

What is a Sum of Geometric Sequence Calculator?

A sum of the geometric sequence calculator is a tool designed 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 (a) of 2 and a common ratio (r) of 3. Our sum of the geometric sequence calculator can quickly find the sum of, say, the first 5 terms of this sequence.

This calculator is useful for students learning about sequences and series, mathematicians, engineers, and anyone dealing with patterns that exhibit exponential growth or decay, like compound interest or radioactive decay over discrete intervals.

Common misconceptions include confusing it with an arithmetic sequence (where terms are added by a constant difference) or thinking it only applies to increasing sequences (it also works for decreasing sequences if the common ratio is between 0 and 1, or alternating if r is negative).

Sum of Geometric Sequence Formula and Mathematical Explanation

The sum of the first n terms of a geometric sequence is denoted as S_n.

The terms of a geometric sequence are: a, ar, ar², ar³, …, ar^n-1.

To find the sum S_n = a + ar + ar² + … + ar^n-1, we can multiply this equation by r:

rS_n = ar + ar² + ar³ + … + arⁿ

Subtracting the second equation from the first:

S_n – rS_n = a – arⁿ

S_n(1 – r) = a(1 – rⁿ)

So, if r ≠ 1, the formula for the sum is:

S_n = a(1 – rⁿ) / (1 – r)

If the common ratio r = 1, then the sequence is a, a, a, …, a, and the sum is simply:

S_n = n * a

Our sum of the geometric sequence calculator uses these formulas.

Variables Explained

Variable	Meaning	Unit	Typical Range
Sn	Sum of the first n terms	Depends on 'a'	Any real number
a	The first term	Varies	Any real number (except 0 for non-trivial sequences)
r	The common ratio	Dimensionless	Any real number
n	The number of terms	Dimensionless	Positive integers (1, 2, 3, …)

Practical Examples (Real-World Use Cases)

Let's see how the sum of the geometric sequence calculator can be used in different scenarios.

Example 1: Savings Growth

Suppose you save $100 in the first month, and each month you manage to save 10% more than the previous month. How much will you have saved in total after 6 months?

• First Term (a) = 100
• Common Ratio (r) = 1 + 10% = 1.1
• Number of Terms (n) = 6

Using the formula S_n = a(1 – rⁿ) / (1 – r):

S₆ = 100(1 – 1.1⁶) / (1 – 1.1) = 100(1 – 1.771561) / (-0.1) = 100(-0.771561) / (-0.1) = 771.561

So, you would have saved $771.56 in total after 6 months. Our sum of the geometric sequence calculator would give you this result instantly.

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 the 5th bounce (i.e., after 4 bounces, considering the initial drop and 4 subsequent downward travels after bounces)?

• First Term (a) = 10 (initial drop)
• Common Ratio (r) = 0.7
• Number of Terms (n) = 5 (initial drop + 4 downward travels after bounces)

The heights it travels downwards are 10, 10*0.7, 10*0.7², 10*0.7³, 10*0.7⁴.

Using the sum of the geometric sequence calculator with a=10, r=0.7, n=5:

S₅ = 10(1 – 0.7⁵) / (1 – 0.7) = 10(1 – 0.16807) / (0.3) = 10(0.83193) / 0.3 = 27.731 meters.

The total distance traveled downwards before the 5th bounce is 27.731 meters.

How to Use This Sum of Geometric Sequence Calculator

1. Enter the First Term (a): Input the initial value of your sequence.
2. Enter the Common Ratio (r): Input the factor by which each term is multiplied. This can be positive, negative, or a fraction.
3. Enter the Number of Terms (n): Input how many terms you want to sum up. This must be a positive integer.
4. View Results: The calculator will automatically display the sum (S_n), the last term (a_n), and rⁿ as you input the values. The table and chart will also update.
5. Interpret the Sum: The "Sum of the First n Terms (S_n)" is the main result, showing the total when you add up the first 'n' terms.
6. Use Reset/Copy: Use "Reset" to clear inputs to defaults and "Copy Results" to copy the main sum and intermediate values.

Understanding the results from the sum of the geometric sequence calculator helps in various fields, from finance (compound interest over discrete periods) to physics.

Key Factors That Affect the Sum of a Geometric Sequence

The sum of a geometric sequence is influenced by three key factors:

• First Term (a): The larger the absolute value of 'a', the larger the magnitude of the sum, assuming other factors are constant. It sets the scale of the sequence.
• Common Ratio (r):
  • If |r| > 1, the terms grow in magnitude, and the sum can become very large (or very negative) quickly as 'n' increases. The sequence diverges.
  • If |r| < 1, the terms decrease in magnitude, and the sum approaches a finite limit as 'n' goes to infinity (see our infinite geometric series calculator). The sequence converges towards zero.
  • If r = 1, the sum is simply n*a.
  • If r = -1, the sum alternates between 'a' and 0 as 'n' increases.
  • If r < -1, the terms alternate in sign and grow in magnitude.
• Number of Terms (n): As 'n' increases, the sum generally increases in magnitude if |r| > 1 or accumulates more terms if |r| < 1. For |r| < 1, the sum gets closer to the sum of the infinite series.
• Sign of 'a' and 'r': The signs of 'a' and 'r' determine the sign of the individual terms and consequently influence the sum. If 'r' is negative, the terms alternate in sign.
• Magnitude of 'r' relative to 1: This is crucial. Whether |r| is greater than, less than, or equal to 1 determines the long-term behavior of the sequence and its sum.
• Value of rⁿ: This term becomes very large for |r| > 1 and large 'n', and very small for |r| < 1 and large 'n', directly impacting the numerator of the sum formula.

Using a sum of the geometric sequence calculator helps visualize how these factors interact.

Frequently Asked Questions (FAQ)

What is a 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).

How do I find the common ratio (r)?

Divide any term by its preceding term. For example, in 3, 6, 12, 24, r = 6/3 = 2 or 12/6 = 2.

Can the common ratio (r) be negative?

Yes, if 'r' is negative, the terms of the sequence will alternate in sign (e.g., 2, -4, 8, -16…). Our sum of the geometric sequence calculator handles negative 'r'.

What if the common ratio (r) is 1?

If r=1, the sequence is constant (a, a, a, …), and the sum S_n = n * a. The calculator accounts for this.

What if the common ratio (r) is 0?

If r=0, the sequence becomes a, 0, 0, 0, … and the sum S_n = a for n >= 1.

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

No, 'n' must be a positive integer representing the number of terms you are summing.

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.

When does a geometric sequence converge?

A geometric sequence converges (terms approach zero) when the absolute value of the common ratio is less than 1 ( |r| < 1 ). If it converges, the sum of the infinite series also converges, which you can explore with an infinite geometric series calculator.

Related Tools and Internal Resources

• Arithmetic Sequence Calculator: Calculates terms and sums for arithmetic sequences.
• Infinite Geometric Series Calculator: Calculates the sum of an infinite geometric series when it converges.
• Compound Interest Calculator: Useful for seeing geometric growth in financial contexts.
• Math Calculators: A collection of various mathematical calculators.
• Financial Calculators: Tools for financial planning and calculations.
• Statistics Calculators: Calculators for statistical analysis.

These tools and resources can provide further insights into sequences, series, and their applications. The sum of the geometric sequence calculator is a fundamental tool in understanding these concepts.