Geometric Series Calculator

First term of the Sequence (a):

Common ratio (r):

Number of terms (n):

Calculate

A geometric series is a type of mathematical series where each term after the first is found by multiplying the previous term by a constant called the common ratio. Geometric series have applications in many areas, including finance, physics, and computer science. Using a Geometric Series Calculator makes it easy to compute the sum of the series and analyze its properties without doing all the manual work.

What is a Geometric Series?

A geometric series is the sum of the terms in a geometric sequence. The general form of a geometric sequence is:

$$a, ar, ar^2, ar^3, ar^4, \dots$$

Where:

• a is the first term,
• r is the common ratio (the factor by which each term is multiplied to get the next term),
• and the terms continue infinitely unless otherwise specified.

The sum of the first n terms of a geometric series can be calculated using the formula:

$$S_n = a \times \frac{1 - r^n}{1 - r} \quad \text{(for } r \neq 1\text{)}$$

Where:

• Sₙ is the sum of the first n terms,
• a is the first term,
• r is the common ratio,
• n is the number of terms in the series.

If the common ratio r is between -1 and 1, the sum of the infinite geometric series (if it exists) is given by:

$$S_\infty = \frac{a}{1 - r}$$

Example of a Geometric Series

Consider a geometric series with the first term a = 3 and a common ratio r = 2:

The series is:
$3 + 6 + 12 + 24 + 48 + \dots$

The sum of the first n terms would be:

$$S_n = 3 \times \frac{1 - 2^n}{1 - 2}$$

For example, if we wanted to find the sum of the first 4 terms, we would plug n = 4 into the formula.

Why Use a Geometric Series Calculator?

A Geometric Series Calculator can help you easily compute:

1. The sum of a geometric series: Whether you’re summing the first few terms or evaluating an infinite series, the calculator will give you the result quickly.
2. The nth term of a geometric sequence: You can input the first term, common ratio, and the term number to calculate any term in the sequence.
3. The convergence of the series: For infinite series, the calculator will check if the series converges (if the common ratio is between -1 and 1) and give you the sum.
4. Visualize the growth: By seeing the terms and sum, you can better understand how the series grows.

How to Use a Geometric Series Calculator

1. Input the first term (a): Enter the value of the first term in the series.
2. Input the common ratio (r): Enter the value of the common ratio.
3. Input the number of terms (n): If you are summing a finite number of terms, enter how many terms you want to sum. For an infinite series, you may not need to input this.
4. Get the result: The calculator will display the sum of the series, the nth term (if needed), and whether the series converges if it is infinite.

Example of Using a Geometric Series Calculator

Suppose we want to find the sum of the first 6 terms of the geometric series where the first term is 5 and the common ratio is 3. By using the formula or a Geometric Series Calculator:

$$S_6 = 5 \times \frac{1 - 3^6}{1 - 3}$$

The calculator would quickly give the result: 3645.

Conclusion

A Geometric Series Calculator is a powerful tool for anyone working with geometric sequences. Whether you’re solving problems in mathematics, finance, or physics, this calculator can simplify the process of finding sums, terms, and analyzing the convergence of geometric series. By entering the first term, common ratio, and the number of terms, you can save time and ensure your calculations are accurate and efficient.