Geometric Sequence Calculator

Our Geometric sequence calculator will evaluate the geometric sequence for the given series with their first term and common ratio. It also supports a wide range of finite and infinite series. 

What is Geometric Sequence?

“The order set of numbers in which every term is multiplied by a constant term to get its next term is known as the geometric sequence”.

Types of Geometric Sequence:

• Finite geometric sequence
• Infinite geometric sequence

Another name for Geometric sequence is known as Geometric Progression. The number that you multiply or divide by each value of the sequence is known as the common ratio. 

  □  Next Term: To get the next term, we have to multiply by the fixed term known as the common ratio.
  □  Previous Term: To get the preceding term, we have to divide the values by the same common ratio. 

Geometric Sequence Formula:

The simple and easy way to find the geometric sequence is to use the geometric sequence calculator. In general, we can write geometric sequences like a, ar, ar^2, ar^3, … Where “a” represents the first term and “r” indicates the common ratio between the terms. The value of r may be positive, negative, or zero.

The general formulas for a geometric sequence are as follows will help you to understand how to find common ratio:

an = a1 * r^(n - 1)

Where:

• an _ nth term of sequences 
• a1 _ first term in the sequence 
• r _ ratio between the terms 

Sum of Finite Geometric Series:

Sn = a (1 - r^n) / (1 - r)

Example:

Suppose a geometric series 3, 6, 12, 24, 48, … with up to 10 terms find. 

• First term of the sequence = 3
• The common ratio = 2

To find the sum of the first 10 terms, we can use the sum of finite geometric series formula:

Sn = a1 * (1 - r^n) / (1 - r)

S10 = 3 * (1 - 2^10) / (1 - 2) 

= 3 * (1 - 1024) / (-1) 

= 3071

Therefore, the sum of the first 10 terms of the sequence is 3071.

Sum of Infinite Geometric Series:

S∞ = a / (1 - r)

Example:

Find the infinite geometric series 1, 1/2, 1/4, 1/8, 1/16, ...

• First term of the sequence = 1 
• Common ratio = 1/2

To find the infinite geometric series, we can use the geometric series calculator but here we substitute the values in the following formula:

S∞ = a / (1 - r)

S = 1 / (1 - 1/2) = 2

Therefore, the infinite geometric series is 2

nth terms of Geometric Series:

an = a1 * r^(n - 1)

Example:

Find the 10th term of the geometric sequence 1, 2, 4, 8, 16, ...

• First term of the sequence = 1
• Common ratio = 2

Use the following formula to find the 10th term:

an = a1 * r^(n - 1)

a10 = 1 * 2^(10 - 1) 

= 1 * 2^9 

= 512

How Geometric Sequence Calculator Operate?

Our geometric series calculator is a tool for anyone who needs to determine the sum of terms regardless of geometric series. Get the instant calculation by putting the below values in the designated fields. 

Data Required:

• Choose the Option: Geometric Sequence, First term, Common Ratio, or Number of Terms
• Insert Values: Put the values according to the point selection 

Result Summary:

• n Terms of the Sequence: Get calculated values for each term in the geometric sequence up to the specified term n.
• Common Ratio (R): Get the amount between each number in the geometric sequence.
• Sum of Sequence: Sum of geometric sequence up to the specified term n.
• Step-by-Step Calculation: Step-by-step breakdown of how each term and the sum were calculated, providing transparency and educational value.

FAQs:

Does a Geometric Sequence Always Increase?

The geometric sequence will not always increase because when it is negative then the next term will be alternating and if their common ratio (r) is in a fraction then the next term will decrease. 

Are Geometric Sequences Always Discrete?

If you have all the terms of sequences in the form of a set then this will be discrete. So in that sense, geometric sequences are the discrete version of exponential functions which are continuous. 

Citations:

Wikipedia: Geometric progression, Elementary properties, Geometric series, Product. 

Cuemath: Geometric Sequence, Examples, Formulas, n^th Term of Geometric Sequence Formula, Recursive Formula, Sum of Finite Formula, Sum of Infinite Formula, Geometric Sequence vs Arithmetic Sequence.