Free Online Geometric Sequence Calculator aid kids to calculate the nth term and the sum of the first n terms of a geometric progression. All you need to provide is an input list of numbers with commas in the respective field and click on the calculate button to obtain the output at a faster pace.
Geometric Sequence Calculator: Wondering how to learn the concept of the geometric sequence and calculate the complicated problems with ease? This is the right page for you as we have discussed GP definition, a geometric sequence formula, how to find the sum of the geometric progression with the common ratio in detailed steps. After referring to the basic information of geometric sequence start utilizing our geometric progression calculator and find the sum of terms of a series with steps in a fraction of seconds.
Geometric Sequences Calculator: If you are struggling to understand what Geometric Sequences are, don't worry as you will find everything related to it here. We included the concept of what a geometric progression is and how to find it manually. You can have detailed steps explaining how to do a summation of Geometric Sequence manually. Our Calculator is quite user friendly and all you need to do is just provide the sequence of numbers in Geometric Progression to avail their summation in no time.
A geometric sequence is the list of numbers where each term in the sequence is multiplied by a constant non-zero number known as the common ratio 'r'. The other name of a geometric sequence is geometric progression.
For instance, 1, 2, 4, 8, 16, 32, ..... is a geometric progression in which every term is multiplied by the common ratio 2 with the prior number in the sequence.
In case the first term of a GP is specified by 'a' and the common ratio between two successive terms is specified by r, then the general form of geometric sequence or progression is as follows:
Finite GP = a, ar, ar2, ar3,.... ,arn-1
Infinite GP = a, ar, ar2, ar3,....,arn-1,.....
Here, arn-1 denotes the nth term of a GP.
The list of geometric sequence formulas is here to help you calculate the various types of problems related to GP like finding nth term, common ratio, the sum of the geometric series:
Sn = a[(rn-1)/(r-1)] if r > 1 and r ≠ 1
Sn = a[(1 – rn)/(1 – r)] if r < 1 and r ≠ 1
Let's consider the geometric series is a, ar, ar2, ar3,.....
Here, the first term is 'a'
The second term is 'ar'
Similarly, the nth term is kn = arn-1
Therefore,
Common Ratio(r) = (Any Term) / (Preceding Term)
= kn/ kn-1
= arn-1/arn-2
The detailed steps that you need to focus & follow while finding the terms of a GP are listed below:
Example:
Find the geometric sequence up to 7 terms if first term(a) = 5, and common ratio(r) = 2.
Solution:
Given a=5, r=2
an = arn-1
a1(first term) = 5x21-1 = 5
a2(second term) = 5x22-1 = 10
a3(third term) = 5x23-1 = 20
a4(fourth term) = 5x24-1 = 40
a5(fifth term) = 5x25-1= 80
a6(sixth term) = 5x26-1 = 160
Hence, the geometric sequence is {5,10,20, 40, 80, 160,...}
Finding the sum of the Geometric sequence can be quite difficult. So, we have come up with simple tricks and steps to solve the finite geometric progression. Jump into the following points and memorize the process of finding the sum of a geometric sequence.
Let’s a, ar, ar2, ar3,....,arn-1 is the given Geometric series or sequence or Finite GP.
Then the sum of finite geometric series is a + ar + ar2 + ar3 +....+ arn-1
The formula to determine the sum of n terms of Geometric sequence is:
Sn = a[(1 – rn)/(1 – r)] if r < 1 and r ≠ 1
Where
a is the first item,
n is the number of terms, and
r is the common ratio.
Also, if the common ratio is 1, then the sum of the Geometric progression is given by: Sn = na if r=1.
Learn the concept of the sum of the terms of GP thoroughly with the help of the provided solved examples. Also, you can assess your knowledge by verifying the answers using SequenceCalculators.com's free online Geometric Sequence Calculator.
Example:
Find the Sum of Geometric Sequence of 10,20,40,80?
Solution:
Given series is 10,20,40,80
At first, we have to find & check the ratio (r) between adjacent members are same or not.
a2 / a1 = 20 / 10 = 2.0
a3 / a2 = 40 / 20 = 2.0
a4 / a3 = 80 / 40 = 2.0
From the above solving, the ratio (r) between every two adjacent members of the series is constant and equal to 2.0
The General Form of a geometric sequence is an = a1 × rn-1
Now, we have to find the Sum of finite geometric series members by using the Geometric Sequence formula:
a + ar + ar2+ ar3+ ar4+ .... + arn-1 = k=0Σn-1 ark = a(1-rn/ 1-r)
The sum of our particular series is as follows:
a ( 1-rn / 1-r) = 10 (1-(2)4 / 1-(2))
= 10 (1-(16) / -1)
= 10 (-15 / -1)
= 10 (15.0)
= 150.0
Finding the nth element
a2 = a1 x r1 = 10 x 21 = 20
a3 = a1 x r2 = 10 x 22 = 40
a4 = a1 x r3 = 10 x 23 = 80
a5 = a1 x r4 = 10 x 24 = 160
a6 = a1 x r5 = 10 x 25 = 320
a7 = a1 x r6 = 10 x 26 = 640
a8 = a1 x r7 = 10 x 27 = 1280
a9 = a1 x r8 = 10 x 28 = 2560
a10 = a1 x r9 = 10 x 29 = 5120
a11 = a1 x r10 = 10 x 210 = 10240
a12 = a1 x r11 = 10 x 211 = 20480
a13 = a1 x r12 = 10 x 212 = 40960
a14 = a1 x r13 = 10 x 213 = 81920
a15 = a1 x r14 = 10 x 214 = 163840
a16 = a1 x r15 = 10 x 215 = 327680
a17 = a1 x r16 = 10 x 216 = 655360
1. What is Geometric Sequence Calculator?
It is a free online tool that aids students to find the terms in a geometric sequence when the first term and the common ratio are known. It only requires the input numbers to calculate the output & display the result in no time.
2. Why use the Geometric Progression Calculator?
By using the Geometric Sequence or Geometric Progression Calculator, we can determine the nth term, common ratio, and the sum of the first n terms of a geometric sequence.
3. How to Use Geometric Series or Sequence Calculator?
The following steps will help you how to use the geometric progression calculator to find a few terms in a geometric sequence:
4. What is a common ratio of geometric sequence?
The number multiplied (or divided) at each stage of a geometric sequence is known as a common ratio.