Geometric Sequence Calculator

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.

General Form of GP or Geometric 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,.... ,ar^n-1
Infinite GP = a, ar, ar², ar³,....,ar^n-1,.....

Here, ar^n-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:

• The general form of GP is a, ar, ar², ar³, etc., where a is the first term and r is the common ratio.
• The nth term of Geometric sequence is a_n = ar^n-1
• Common ratio (r) = a_n/ a_n-1
• The geometric sequence formula to determine the sum of the first n terms of a Geometric progression is given by:

S_n = a[(r^n-1)/(r-1)] if r > 1 and r ≠ 1

S_n = a[(1 – rⁿ)/(1 – r)] if r < 1 and r ≠ 1

• The nth item at the end of GP, the last item is l, and the common ratio is r = l / [r (n – 1)].
• The sum of infinite series, that is the sum of Geometric Sequence with infinite terms is S∞ = a / (1-r) such that 1 >r >0.
• If there are 3 values in Geometric Progression, then the middle one is known as the geometric mean of the other two items.
• If a, b, and c are three values in the Geometric Sequence, then “b” is the geometric mean of “c” and “a”. This can be written as b = √ac or b² = ac
• Assume that “r” and “a” are the common ratio and first term of a finite geometric sequence with n terms. Therefore, the kth item at the end of the geometric series will be ar^n-k.

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 k_n = ar^n-1

Therefore,

Common Ratio(r) = (Any Term) / (Preceding Term)

= k_n/ k_n-1 

= ar^n-1/ar^n-2

How to Find the Terms of Geometric Progression? 

The detailed steps that you need to focus & follow while finding the terms of a GP are listed below:

• Firstly, take the first term 'a'.
• To find the second term, multiply 'a' with the common ratio 'r'; a × r.
• Similarly, repeat the same process. Keep multiplying the common ratio with the prior term & find the required number of terms. 
• The other way to find the various terms in a GP is by substituting the value of n in ar^n-1.
• Finally, you have seen two ways to find the terms of GP. Pick any of them and solve the problems of geometric sequence effortlessly. 

Solved Example on Geometric Sequence

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

a_n = ar^n-1

a₁(first term) = 5x2^1-1 = 5

a₂(second term) = 5x2^2-1 = 10

a₃(third term) = 5x2^3-1 = 20

a₄(fourth term) = 5x2^4-1 = 40

a₅(fifth term) = 5x2^5-1= 80

a₆(sixth term) = 5x2^6-1 = 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,....,ar^n-1 is the given Geometric series or sequence or Finite GP.

Then the sum of finite geometric series is a + ar + ar² + ar³ +....+ ar^n-1

The formula to determine the sum of n terms of Geometric sequence is: 

S_n = 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: S_n = 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 a_n = a₁ × r^n-1

Now, we have to find the Sum of finite geometric series members by using the Geometric Sequence formula: 

a + ar + ar²+ ar³+ ar⁴+ .... + ar^n-1 = _k=0Σ^n-1 ar^k = 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

Example for Finding nth term of Geometric Sequence

Example for Finding Sum of n terms of Geometric Sequence